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A theory of generalized functions based on holomorphic semi-groupsCitation for published version (APA):Graaf, de, J. (1979). A theory of generalized functions based on holomorphic semi-groups. (EUT report. WSK,Dept. of Mathematics and Computing Science; Vol. 79-WSK-02). Technische Hogeschool Eindhoven.
Document status and date:Published: 01/01/1979
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TECHNISCHE HOGESCHOOL EINDHOVEN
NEDERLAND
ONDERAFDELING DE~ WISKUNDE
TECHNOLOGICAL UNIVERSITY EINDHOVEN
THE NETHERLANDS
DEPARTMENT OF MATEEMATICS
A THEORY OF GENERALIZED FUNCTIONS BASED ON
HOLOMORPHIC SEMI-GROUPS
by
J. de Graaf
T.H.-Report 79-wSK-02
March 1979
Contents
Abstract
Chapter O. Introduction
Chapter 1. The space SX,A
Chapter 2. The space TX,A
Chapter 3. The pairing of SX,A and TX,A
Chapter 4. Characterization of continuous linear mappings between
the spaces SX,A ' SX,A' Sy,B and SY,B
Chapter 5. Topological tensor products of spaces of type SX,A ' SX,A
Chapter 6. Kernel theorems
Appendix A. Functions of self-adjoint operators
Appendix B. Holomorphic semi-groups
Appendix C. A Banach-Steinhaus theorem
Acknowledgement
References
1
2
12
26
33
39
48
59
65
67
69
70
71
Abstract
In a Hilbert space X consider the evolution equation
du dt = - At:!
wit,h A a non-negative unbounded self-adjoint operator. A is the infinite
simal generator of a holomorphic semi-group. Solutions u(') : (0,00) + X of this equation are called trajectories. Such a trajectory mayor may
not correspond to an "initial condition at t = 0" in X. The set I
of trajectories is considered as a space of generalized functions. The
= U e -At (X) • test function space is defined to be SX,A t>O
For the spaces SX,A,SX,A I discuss a pairing,topologies, morphisms,
tensor products and Kernel theorems. Examples are given.
AMS Classifications 46F05 46F10
- 2 -
CHAPTER O. Introduction
In a very inspiring paper, De Bruijn [B] has proposed a theory of
generalized functions based on a specific one-parameter semig~oup
of smoothing operators. This semi group constitutes what is known as
a holomorphic semigroup of self-adjoint bounded operators on
L2 (lR) •
This observation has enabled me to generalize De Bruijn's theory
and to place it in a wider context of functional analysis.
By this approach, we can introduce De Bruijn type generalized functions
and corresponding spaces of test functions on manifolds and
also on duals of nuclear spaces. We can construct the spaces of
test functions even so as to be invariant under one or more given
operators. In [B] such a given operator is the Fourier transform.
Even so, there are many possibilities, allowing us to impose further
conditions. In the sequel I study extensively the spaces of test
functions and distributions that arise in this fashion. I characterize
their topologies and morphisms. Moreover, I give necessary and sufficient
conditions for the validity of Kernel Theorems.
In this introduction I illustrate the ideas by a few examples. The first
example is very simple and I discuss it in some detail. I use it to give
an outline of the general theory. The remaining examples merely
illustrate how to apply the general theory; they are not worked out
in detail. They could be the subject of further investigation.
- 3 -
Example 1
Let us consider the elementary diffusion equation
(1) X E JR, t > 0 •
A classical solution F(x,t) with the property Vt > 0 F(-,t) E L2
(JR)
will be called a trajectory. Corresponding to any "initial condition"
f E L2
(JR) there is a trajectory given by the well-known formula
F(x,t)
00 ... 2
J f(~)exp(- (x - ~) )d~
4t t > 0 .
From this expression it follows that the special trajectory FC-,t) Mtf,
f E L2 (JR) has the properties
(i) Vt > 0 VT > 0 MtF(.,T) = F(', T + t) = MTF(·,t) •
(ii) Vt > 0 F(x,t) is an entire analytic function of x.
With the aid of the maximum principle for parabolic equations, [PW] p.160,
it follows that (i) is true for ~ trajectory and hence also (ii) is
true. From the positivity and linearity of the integral operator Mt it
follows that for a given trajectory F(·,t) there exists at most one
g E L2 (JR) such that F(',t) = Mtg. In general no such g does exist at
all, consider e.g.
G(x,t) 2
-~ (x - a) 2(nt} exp - 4t
Note that G(x,t) "tends" to the a-function a (x - a) for t '" O. Any
derivative of G is again a trajectory.
Following De Bruijn we gather all trajectories of (1) in a complex
vector space S' . The elements of $' are to be interpreted as generalized
functions. As we just have seen, a generalized function sometimes
corresponds to an element of L2 (JR) and, if so, the correspondence is
unique. The test function space $ is defined to be the dense linear
subspace of L2 (JR) consisting of smooth elements of the form Mth,
h E L2 (JR), t > O. The densely defined inverse of Mt is denoted by
M . For each (jJ E S there exists T > 0 such that M Q) makes sense. -t -T
The 2airing between Sand $' is defined by
- 4 -
(2) cp € 5 , F E S'
Here (. I .) denotes the inner product in L2 (lR) •
The definition makes sense for e positive and sufficiently small.
Since Mt is a symmetric operator and MtMT
= Mt +T
, the definition (2)
does not depend on the specific choice of e. In the following chapters
we provide S with suitable topologies and we prove
(i) For each fixed F E 5' the map,» <"F> is a continuous linear
functional on S. (ii) For each continuous linear functional t on 5 there exists exactly
one trajectory F~ such that for all cp € 5 we have i(cp) = <cp,Ft>.
Suppose P is a densely defined linear operator in L2 (lR) with a densely
defined adjoint P* which leaves 5 invariant, i.e. P*(5) c 5. Then P can
be extended to a continuous mapping from 5' into itself by
* <~,PF> = <P cp,F> •
Examples of such operators Pare M , R , T , Z" V and compositions of . a a a 1\ ~ax these. Here (R f) (x) = e f (x), (Tbf) (x) = f (x + b), (Z). f) (x) = f eX. x) I
df(xT . (Of) (x):= dx w~th a,b,A € ]R.
Finally we want to show that certain strongly divergent Fourier integrals
can be interpreted as elements of 5' •
This interpretation is closely related to the Gauss-Weierstrass summation
method. Let g be a measurable function on ]R such that for each e > 0 the -ey2
function g(y)e
J:ro
g(y)eiyxdy can
is given by G(x,t)
is in Ll (lR) • The possibly divergent Fourier integral
be considered as an element of 5'. The trajectory G
f oo -ty2 iyx := _00 g(y)e e dy. This simpe illustration of
our theory certainly has some eleg.ant features, but in some respects
it is too simple. Since Mt is not a Hilbert-Schmidt operator there is
no Kernel Theorem in this case. That is to say, there exist continuous
linear mappings from S into S' which do not arise from a trajectory of
the diffusion equation in ]R2. Cf. Chapter 6 case b.
- 5 -
Partial sketch of the general theory
We consider an evolution equation in a Hilbert space H
(3) du dt :::: -Au
Here A is a densely defined unbounded operator such that the exponential -At
e can be given a meaning. A trajectory of (3) is defined to be a
mapping F : (0,00) ~ H with the property Vt > 0 V, > 0 e-A'F(t) = F{t + T).
Examples of trajectories are F(t) = e-Ath, h € H.
Further we introduce a test function snace S = U e-At(H). ... t>O
In order to make all this into a non-trivial interesting theory of
generalized functions which generalizes the situation of example 1, we
must meet some requirements. -At
(i) -A must be an infinitesimal generator. In other words e must
have a meaning and it is an everywhere defined bounded operator.
-At (ii) e must be injective for each t > o.
At -At -At (iii) The inverse e of e has to be unbounded in order that e is
"smoothing".
These conditions are fulfilled if -A is the infinitesimal generator of a -At
one-parameter holomorphic semi-group e t ~ O. Here holomorphic
means holomorphic in t (as a bounded operator-valued function). This is
not related to any kind of smoothness or analyticity of the "test functions"
in S. The next chapters deal with the case that A is a non-negative
unbounded self-adjoint operator in a separable Hilbert space H. In example
1 we had H = L2 OR), A = - d2
2 with domain D(A) = H2 (JR) , the 2
nd order
Sobolev space. dx
Example 2
ex > 0
In this case
with
Kt(aix,y)
- 6 -
00
-At J (e u) (x) = Kt (cqX,y)u(y)dy
-00
00
! J exp(-lkI2at)cos k(x -y)dk •
o
For a = 1 the kernel Kt has been calculated in example 2.
For a = ~ we have
1 t Kt(~ix,y) =; 2 2
t + (x - y)
which is just the Poissonkernel for solving the Dirichlet problem in a
half plane t ~ O. This is not surprising, since, at least formally,
The following properties of the corresponding test function spaces S are
easy to verify
a > ~ S consists of entire analytic functions.
S consists of functions which are analytic in a strip around the
real axis. 00
a < a < l:1: The elements in S are C but non-analytic in general.
Finally we wan.t to show that for each a > 0 certain strongly divergent
Fourier integrals can be interpreted as elements of S·. Cf. Example 1.
Let g be a measurable function on lR such that for each £ > 0 the function'
1 12a foo iyx g(y)exp - (;: Y is in Ll (JR) • The possibly divergent integral _00 g(y)e dy
is in S·, The trajectory G is given by
G(x,t)
-00
For a = l:1 this corresponds to Abel's summation method.
- 7 -
Example 3
H ::: L2 ([ 0 , 2 11 J ) I Cl > 0
In this case
with
For Cl 1
u(O) == U(21T),
21T
-At f (e u) (x) == Kt
(Cl; x,y) u (y) dy
1 21T
oo
n==-""
o
exp{- Inl2Cl
t + in(x - y)} =
u· (0)
=_1 {1+2 21T I 2Cl
exp{-n t}cos n(x - y) • n=l
1 -t Kt(lix,y) = 2; 83 (x - y,e )
Here 83
is one of Jacobi's theta functions [WW] p. 464.
1 sinh t 21T cosh t - cos (x - ~) •
U'(21T)}.
In the latter case the smoothing integral operator is the Poisson kernel
for the solution of the Dirichlet problem for the unit disc. For a
general discussion of this phenomenon see Example 5. 2 -At
For each Cl > 0 Kt(a j o,') E L2
([0,21T] ). In other words e is a Hilbert-
Schmidt operator. This implies that for generalized functions of this type
a Kernel Theorem holds true. See Chapter 6. Finally, we remark that
certain strongly divergent Fourier series do have a meaning within this
theory. oo
Let a > 0 be fixed. Consider a sequence of complex numbers {c } such n n=-""
that {Ic lexp - E\nI2Cl
} is a bounded sequence for each E > O. Then n. 00 l.nx
E c e can be viewed as a generalized function. The corresponding n-- oo n trajectory is given by G(x,t) ::: roo c exp{- Inl 2Clt + inx}.
n=-oo n
- 8 -
Example 4 1 1
H d 2 2(1 2 2S
A = (- -2) + (x) , (1 > 0, B > 0 , dx
1 ex
D (A) = H OR)
here F is the Fourier operator. -At For ex = e = ~ an explicit representation of e. by means of an integral
operator can be found in [B].
This special case has been extensively investigated by De Bruijn and
Janssen [BJ, [JJ. A part of our results, especially in chapters 3 and 4
€onsists of generalizations and adaptations of their work.
Whenever ex a the test function space is invariant under Fourier .. transformation. We conjecture that the Gelfand-Sllov test function classes B S , [GSJ Vol. II Ch. 4 are the same as the test function spaces 5 of 0.
this example.
The preceding examples can all be generalized to lR n in a direct obvious way.
These generalizations can also be obtained by application of the theory of
Chapter 5 where topological tensor products of the spaces 5 and 5' are
formed. Repeated formation of tensor products of the Sand 5' spaces of
example 3 leads to distribution theories on n-dimensional tori. Our type
of distributions can be introduced on any, not too bad, differentiable
manifold M by taking for A a positive elliptic differential operator which
is self-adjoint in an L2-space over M. On a compact Riemannian manifold
one may take the Laplace-Beltrami operator. This can be done on the
q-dimensional unit sphere in lR q+1 for example. In the latter case how-
ever a very nice semi-group of smoothing operators can be chosen which
is closely related to the Poisson-integral solution of the Dirichlet
problem for the unit ball in lR q+1• This is the subject of the last
example.
- 9 -
Example 5
(4 )
Here ilLB denotes the Laplace-Beltrami operator, 1 denotes the identity
operator.
After introducing orthogonal spherical coordinates xi = rF. (91
, ••• ,6 ), . q+1 ~ q
1 :::; i :.;; q + 1 ~n lR we obtain for the Laplacian
(5) q+1
f:, = I i=l
a Cl 1 .... _+-f:, r ar 2 LB r
From this it follows, see [M] p. 4, that an m-th order spherical hal:oonic
is an eigenvector of ilLB with eigenvalue equal to -m(m + q - 1). Then a
simple calculation shows that each m-th order spherical harmonic is an
eigenvector with eigenvalue m of our operator A. -t
Introduction of r = e in (5) transforms the Laplacian into
2t 32 a
6. '" e {- - (q - 1)- + 6.LB
} • at2 at
The expression between { } can be factored into two evolution equations.
Thus
2 ~ ~(q - 1)1 + {~(q - 1) 1 - lI
LB} ]
[Jl - ~(q - 1)1 + {~(q - 1)2 - II }~J at LB
3 The second factor can be written as at + A I with the A from (4).
From these considerations it follows that substitution of r = e-t
in
the Poisson integral formula, cf. [M] p.41, leads to an integral ex--At
pression for e •
- 10 -
(6)
Here t;. and T1 denote elements of sq, 1. e. unit vectors in lR q+ 1. The
"surface measure" of sq is denoted by dw , while F;.'n denotes the inner q
product of t;. and n. For fixed n the kernel in (6) denotes the trajectory of the o-function
centered at n. For a representation of the kernel involving zonal harmonics. See -At
[G Se]. Here the operator e is obviously Hilbert-Schmidt. Then from
Chapter 6 it f 11 th t f th di o ows a or e stribution theory arising from the operator
A in (4) Kernel Theorems are available. Similar to example 3 certain
strongly divergent sequences of spherical harmonics can be "summed".
I conclude this introductory chapter by summarizing briefly the contents
of the next chapters.
In Chapter I the test function space SX,A is discussed. Here X is a
separable Hilbert space, A is a non-negative unbounded self-adjoint
operator in X. SX,A is provided with a non-strict inductive limit
topology. A base of the neighbourhood system of 0, is given. Bounded and
compact sets are characterized. A necessary and sufficient condition for
the convergence of a sequence is given. In Theorem 1.11 the properties
of SX,A are summarized using the standard terminology of the theory on
locally convex topological vector spaces.
In Chapter 2 the program of Chapter 1 is repeated for the space of
trajectories T X,A' A characterization of the trajectories is given.
In Chapter 3 the pairing between elements of SX,A and TX,A is discussed.
Weak topologies are introduced. Banach-Steinhaus theorems are proved.
Strong and weak convergence of sequences are compared. It is proved
that SX,A and TX,A are both strong and weak duals of each
This motivates the choice of the notation TX,A = SX,A' In Chapter 4 continuous linear mappings between the spaces
SY,B and SY,B are extensively discussed and characterized.
of extending linear operators is considered,
other.
SX,A' SX,A' The problem
Chapter 5 deals with completions of topological tensor products of the
spaces of Chapter 4.
- 11 -
Chapter 6 is devoted to Kernel Theorems. We give necessary and sufficient
conditions on A and B in order that all linear mappings of Chapter 4 are
themselves element of spaces SX®Y,A EE1B ' etc.
- 12 -
CHAPTER 1. The space SX,A
In a complex separable Hilbert space X with inner product (.,o}X we
consider an unbounded non-negative self-adjoint operator A. This operator
will be fixed throughout chapters 1,2,3. Since A is self-adjoint it
admits a spectral resolution
For details and conventions on such spectral resolutions we refer to
Appendix A •. We define
-At e
o
00
tEa: •
For t E lR the operator Mt is self-adjoint, Mt is bounded iff Re t ~ O. -1
Mt is unitary iff Re t O. Further Mt is invertible and 'It E a: Mt M_t + On a: , i.e. the set of tEa: with Re t > 0, the operators Mt establish a
one-parameter semi-group of bounded injective operators on X. For more
details on such holomorphic semi-groups see Appendix B.
We now introduce our space of test functions SX,Ao
Definition 1.1.
SXA= U {MtXlxEX}= U Mt(X) , Re t>O Re t>O
SX,A is a dense linear subspace of X. From the semi-group properties the
following equalities immediately follow for any 0 > 0
Each of the spaces M (X), t > 0, can be considered as a Hilbert space. t
The inner product is
(u,V)M (X) t
- 13 -
The completeness of Mt(X) follows from the closedness of M_to
Mt(X) consists of exactly those u E X for which 00
< 00 •
a
If locally no confusion is likely to arise we suppress as many subscripts
as possible. In chapters 1, 2 and 3 we shall write consistently: 5 for
SX,A' Xt for Mt(X), (0,0)0 or (',.) for (",)X'
Defini tion 1. 2.
The strong topology on S is the finest locally convex topology on S for
which the natural injections it Xt
+ 5, t > 0, are all continuous.
In other words S is made into a locally convex topological vector space
by imposing the inductive limit topology with respect to the family
{Xt}t>O • Cf. [SCHJ Ch. 11.6. Since the natural injections XL + Xt T > t > a, are all continuous the inductive limit topology is already
brought about by the family {X1
/ n } nEJN' A subset U c S is open iff for
-1 every t > 0 the set it (U) = U n X
t is open in X or, equivalently, iff
for every n E IN the set U n X1
/ n is open in Xl/n0 The inductive limit
here is more complicated than in the case of the Schwarz test function
spaces because our inductive limit is not strict! A closed subset of X
when considered as a set in Xt
, a < t < ., is not necessarily closed.
We shall see that open sets in 5 are always unbounded.
We introduce some notations:
- B denotes the set of everywhere finite real valued Borel functions -0
on lR such that Yt > 0 the function ~(x)e is bounded on [0,(0)'
- B c B contains those ~ with ~(x) > +
a Yx ~ 0
T c B consists of those ~ which are zero for x < a and piece-wise +
constant on the intervals [O,lJ, (n,n + 1J, n E IN. The value of ~
on such an interval is a natural number.
- B (A) , B (A), T (A) denote the corresponding sets of self-adjoint +
operators in X defined by
1"
- 14 -
00
A foo 2 The domain of ~( ) consists of exactly those x € X with 0 ~ (A)d(EAx,x)
It follows that S is contained in the domain of each operator ~(A). -At
Further: V~ € B Vt > 0 the operators ~(A)e = ~(A)Mt are bounded
and self-adjoint. The operators in B+(A) are all strictly positive.
Definition 1. 3.
For each ~ € B we introduce the following semi-norm (which is actually +
a norm)
= (~(A)u,u)~ = {J o
Further we define the sets
U € S •
u = {u € S I (~(A) u, u) <: E:2 , ~ € B, E: > O} •
~,E: +
The next theorem is very fundamental.
Theorem 1.4.
I. V~ E B V£ > a U is a convex, balanced, absorbing open set in + ~,E:
the strong topology. In other words the semi-norms p are continuous.
II. Let a convex set Q c S be such that for each t > 0 Q n Xt
contains a
neighbourhood of 0 in X t' Then n contains a set U with ~ € Band £ > O.
~,£ +
III. That set of U,', ' ~ E T, £ > 0, establishes a base of the neighbourhood ,/"E:
system of 0 in S. In other words the strong topology in S is generated
by the sets U ~,E:
IV. For any open set V c S , 0 E V and for each t > 0 the intersection
Xt
n V is an unbounded set in Xt
•
< 00 •
- 15 -
Proof
I. A standard inner product argument shows that ~I. is convex, balanced 'f,e:
and absorbing. For the terminology see [yJ Ch. I.1. We now show that
for each t > 0 U~,£ n Xt is an open set in Xt
•
By definition
2 U n Xt = {ulu EXt' (lJ!(A)u,u) < E } ~/£
2 Because of (ljJ(A)u,u) '" (1jJ(A)M2t
M_t
u,M_t
u) ::; 1IlJ!(A)M2t
IlIlM_t
ull ""
= 11lJ! (A) M2tllil u II~ and the boundedness of ljJ (A) M2t the sesquilinear form
(lJ!(A)u,u) is continuous on Xt • Therefore the set of u's in Xt
for which 2
(lJ!(A)u/u) < E is open in Xt*
II. We proceed in six steps
a) Introduce the projection operator P = n
the radius of the largest open ball in IN. Let r be
n,t
Thus
r n,t
[u E ~ n P (X) n X J} n t
b) For t > T ~ 0 we have e(n-l) (t-')r n,.
The proof of this runs as follows
n
n
fits in ~ n P (X ). n t
I 2At 2 e d(EAu,u) < p ] ~
n-1
.",. n (t-.) '" r ::; e r n,t n,c
II P u 112 n • .f
2AT -2 (n-l) (t-"r) e d(EAu/u)::; e
n-l n-l
Therefore if we take u such that
- 16 -
II P u 1\ n t (n-l) (t--r)
< e r (n-1) tt--r) n,T
it follows that II Pull < r • Consequently ,n l' n,T
r n,t
~ e r. The n,t
proof of the other inequality runs similarly,
00 p -1 c) For any fixed p > 0 and t > 0 the sequence E -1 n (r t) is convergen=. n- n,
Indeed, there exists an open ball in X , l' < t, with sufficiently small l'
radius E and centered at 0 which lies entirely within Q n Xt
• Then
Vn E ID r ~ E and with the first inequality in b) it follows that -1 n,T -1 - (n-1) (t--r) -1 - (n-1) (t-t)
(r t) ::; (r ) e ::; E e • n, n, T
d) We define a function X on ]R by
x(x) n E: ID
x(O) = X(~)
x(x) = 0 for x < 0
-EX and claim that X E B • For the proof we have to verify that X(x)e is
+ bounded on [0,00) for every E > O. On (n-l,n] and for every 1', 0 S T S 1
applying b) with t = 1 we find
-EX 4 2n-£(n-l) -2 X (x) e s 4n e (r 1) S n,
< 4 4 2+(n-l) (21'-E) ( )-2. - n e r n,T
Taking l' < ~E and invoking c) the result follows.
e) We now prove
(x (A) u, u) < 1 .. U E Q •
00 2 Suppose u E X
t for some t > O. Then E II Pull < 00 and for some 1',
n=l n t o <: , < t it follows with b)
(ft )
Further because of our assumption (*) and b)
II Pull n , n, -2 n,-n -2
sell Pnu 110 < ~n r 1 e S ~n r n, n, T
Therefore 2n2 p u E Q n X n T
for every n E ID.
- 17 -
In X we represent u by l'
N I;X)
I 1 2 I 1 u = 2n2
(2n P u) + ( -)u n=l n n=N 2n2 N
with
00
'I _1_)-1 'I P L L u. j=N 2j2 n=N n
With (*) we calculate
Since n n X contains an open neighbourhood of a for N sufficiently large T
u E Q n X . N .-
Finally we gather that u is a sub-convex combination of elements in
n n X which is a convex set • .-(Aposteriori it is clear that u E n n X
t).
f) We finish the proof of II by taking $ E T such that on (n-1,nJ it is 4 2n -2 equal to the smallest natural number greater than 4n e (r 1) •
n,
III) Any convex open neighbourhood of a satisfies the conditions of the set Q in II.
Therefore it contains a set V,I, with 1ji € T and £ > a. '/"IE:
IV) V contains a set V,I, • The results lIb and lIe show that V,I. n X contains ,/",£ ,+,,£ t
arbi trary large elements. [J
Remark. Similar to the proof of part lone proves that the sets
are closed.
Definition 1.5.
2 s €: €: > a}
A subset W c S is called bounded if for each a-neighbourhood U in S there
exi~ts a complex nunilier A such that W C AU • Cf. [SeB] 1.5.
- 18 -
In the next theorem we characterize bounded sets in S.
Theorem 1.6.
A subset W c S is bounded iff
3t > 0 3M > 0 Vu E: W II u II :s; M . t
Proof.
W is bounded iff V~ E: B+ 3M ~ 0 Vu E: W (~(A}u,u) ~ M. Therefore by taking
~ = 1 we observe that W is a bounded set in X. SO we have
3p > 0 Vu E W (u,u) :s; p
and
M
f 2At 2Mt
(*) Vu E: W VM > 0 Vt > 0 e d(tAu,U):S; e p
o
~) Vu E: W (~(A)u,u) = (~(A)M~ M u,M u) "s -s -s
which is a constant only depending on ~ •
• ) Suppose the statement were not true then
Vt > 0 \1M > 0 3u E W II Mull > M • -t
By induction we define two sequences of real numbers {t }, {N }, t + 0, N + 00 as n n n n
n ~ 00 and a sequence {u } c W as follows n
n = 1
n '" 9. + 1
= 2. Then take Nl > 0 and u 1 E W such that
> 1 •
Suppose Vt ~ ~t9. Vu E: W VK > 0 2tA
e d(tAU/U) ~ 9. + 1 is true.
Then W is a bounded set in Xt
, t $ ~t , because with the aid of (*) 9.
00 00
f 2At
e d(EAu,u) J 2+N JI'p
$ e + £ + 1 •
o o
- 19 -
If our sequences terminate for a certain value of n then W is a bounded set.
If not, define the function X(A) on (a,"") by
2At n neAl = e on the interval (N 1,NJ n- n
-At Since n(A)e is a bounded function for t > 0, nEB
n = 1 ,2, • •• •
+ and the sequence
{(n(A)u ,U )} should be bounded. However (n(A)u iU ) = n n n n f~ n(A)d(E,u ,u ) > n + 1.
I\. n n Contradiction!
In the next theorem we give a characterization for the convergence of
sequences {u } in the strong topology of S. n
Theorem 1.7.
Proof.
u + 0 in the strong topology of S iff n
3t > C {u } c X and II u II -+ 0 • n tnt
~) For any ~ E B+ we have (~(A)u ,u ) = (~(A)M2tM t U 1M u) n n - n -t n ::; II1/! (A) M2tllil Ai u II 2
-t n + 0 as n + 00 because ~(A)M2t is a bounded operator on X.
-*) Suppose u . .,. O. Then VI/! E B+ (1/;(A)u ,u) .. O • n n n We conclude that the sequence {u } is a bounded set. So 38 > 0 'In E :IN lIu II n n T
::;
Further, taking ~ 1 , it is clear that lIunl~ + 0 as n + "". From this we derive
+ 0 as n -+ "".
Now we show II un I~ -r 0 for any t < 1'.
A.
- 20 -
The second integral can be estimated unifo::mly
= f -2A(T-t) 2ATd (E ) _< e e ,u ,U 1\ n n
L L
00
e -2L (-r-t) J L
2AT . _< -2L(T-t) 2 e d (E, u ,U) e e.
1\ n n
By taking L sufficiently large the second term in (,1) can be made smaller
than ~~ uniformly in n.
The first term in (*) tends to 0 as n + 00 because of (*).
Theorem 1.8.
I. Suppose {u } is a Cauchy sequence in the strong topology of S. Then n
3t > a {u } c Xt
and {u } is a Cauchy sequence in X • n n t
II. S is sequentially complete, i.e. every cauchy sequence converges to a limit
point.
Proof
I. An argument similar to the proof of the preceding theorem.
II. Follows from I and the completeness of Xt •
Next we characterize compact sets in S.
Theorem 1.9.
A subset K c S is compact iff
3t > 0 K c Xt
and K is compact in Xt
•
Proof.
<=) Let {rl } be an open covering of K in S. Then {n n Xt
} is an open u u
covering of K in Xt
• Since K is supposed to beN compact in Xt there exists
a finite subcovering N
{rl }, 1 sis N, with i;;1 (n n X ) :> K. But ai
ai
t
then certainly i~1 rl a ,:> K. ~
~) Since K is c~mpact it is bounded and therefore it is a bounded set with
[
- 21 -
bound e in XT
for some T > O. We show that K is compact in Xt
whenever
t < T. Consider a sequence {u } c K. There exists a converging sub-n
sequence {u } with u ~ u ~ K as j + ~, convergence in S. Then also nj nj
u ~ u in X-sense. Put u nj nj
- u = vj
• {vj
} is a bounded sequence in
Xi and IIVj II ~ 0 as j ~ 00. Now we find ourselves in exactly the same
posi tion as in the proof of the only-if-part of Theorem 1.7. We conclude
II Vj lit + 0 whenever 0 s; t < T. OUr subsequence {un} converges to u in
Xt-sense. This shows the compactness of K in Xt
jfor 0 s; t < T. 0
Lemma 1.10.
Let u E X be such that u ~ D(w(A» for all WEB. Then u E S. +
Proof d _
If not, then for every t > 0 the integral f~ e2At
d(EAu,u) is divergent.
Pick sequences of real numbers {til, {Ni
} with ti ~O and Ni t ~ such that
for all i E ]\j
N i 2A
( t.
. e ~d(EAu,u) > 1 • . N. 1 3.-
At i Define W on (0,00) by W(A) = e
2 Joo 2 111fi (A) u II = 0 1jJ (A) d (E AU, u) = 00
Con tr adi cti on!
on
In order to make a link with the literature on topological vector spaces we
now describe the properties of our space S by using the standard terminology
of topological vector spaces. [SCH).
The terminology is explained in the proof.
Theorem 1.11
I SX,A is complete
II SX,A is bornological
o
- 22 -
III. SXtA is barreled
IV.
V.
Proof
SX,A is Montel iff for every t > 0 the operator Mt
as an operator on X.
SX,A is Nuclear iff for every t > a the operator Mt
HS (=Hilbert-Schmidt) operator on X.
.. At e is compact
-At = e is a
I. Let {x } be a Cauchy net. The a's belong to a directed set D. For each ~
neighbourhood rt :3 0 there is Y E D such that whenever a >-- y and S >- y
one has xa - Xs ~ rt. We now prove that there exists an XES such that
x ~ x in the strong topology. Let x be the limit of x in X-sense, a a For each tjJ E B the net {tjJ(A)x} converges in X-sense to a limit x,/,,
+ a 'i'
Because of the closedness of tjJ(A) one has xtjJ E D(tjJ(A» and xtjJ = tjJ(A)x.
The result follows by applying Lemma 1.10.
II. Every circled convex subset n c S that absorbs every bounded subset W c S
III.
has to be a neighbourhood of O. Let Bt
be the open unit ball in Xt
, t > O.
Bt
is bounded in S therefore for some E: > o one has E:Bt
c rt f1 Xt
•
We conclude that for every t > 0 the set n f1 Xt
contains an open neighbourhood
of O.
But then according to Theorem 1.4.II rt contains a set U,/, • 'i',E:
A barrel V is a subset which is radical, convex, circled and closed. We
have to prove that every barrel contains an open neighbourhood of the
origin. Because of the definition of the inductive topology V '1 Xt
has
to be a barrel in Xt in the Xt-topology, for every t > O. Since Xt is a
Hilbert space there exists an open neighbourhood of the origin [ with
[ c V n Xt ' Again the conditions of Theorem 1.4.1I are satisfied so that
V con tains a se t iJ", • 'i"€
IV. We have to prove that every closed and bounded subset of S is compact iff
for every t > a the operator Mt
is compact.
~) Suppose Mt is compact for every t > O. Let W be a closed and bounded
subset in S. Then W c Xt
for some t > 0 and W is closed and bounded
in Xt , See'I'heorelll 1.6. Take 1",0 < 1" < t. We claim that W is a compact
set in X , To see this, consider the following commutative diagram where C; T
v.
- 23 -
denotes the natural injection
Xt c: X
T
M' i i M t T
Xo ,.. Xo M t-T
The vertical arrows are isomorphisms. So c:;. is a compact map and W
compact in X . Consider an open covering {C } of W in 5, then {C T a a
is an open covering of W in X • Because of the compactness of W in T
there is a finite subcovering:
W c:
N u
i=l (C
a. ~
n X ) • T
N But then certainly W c: u C , which shows the compactness of W.
i=l a i
is
n X } T
X T
~) Suppose S is Montel, i.e. each closed and bounded set is compact. Let
{u } be a bounded sequence in X. Pick any fixed t > O. Consider the n
.. )
sequence {M u }. Consider the closure in S of this sequence. This closure t n
is a bounded set and, according to our assumption, compact. Thus
{Mtun
} contains a converging subsequence in S: Mtu ~ v. This sequence nj
certainly converges in X. SO Mt
must be a compact operator •
We
a)
proceed in four steps: a,b,c and d.
Suppose that for every t > 0 Mt
is H.S., i.e. for any orthonormal
basis {e.} in X ~
00
Vt > 0 I II Mte.112 < 00.
i=l ~
It follows that M t
-At e is compact. Hence A has a purely discrete
spectrum with no finite accumulation point and all eigenspaces are
finite dimensional. We order the eigenvalues in a non-decreasing
sequence in which each eigenvalue occurs as many times as the
b)
- 24 -
dimension of its eiger.space
A -+ 00 n as n -+ 00 •
We choose an orthonormal base of eigenvectors {e }. Together with 00 - 2Ak t n 00
(*) we obtain for each t > 0 Lk=l e < 00 and aposteriori L
k=l
Let B1 and B2 be Banach spaces. A bounded operator T lZalled nuclear if there exists a sequence {f'} C 8' ,
n - 1 {a } c 82 and a sequence of complex numbers {c } such ~n
and
Our
sup (II f' II, II g II) < 00
n n
I Ie i < ro
n=l n
Vx E
operator
-At e :u
B1
e
m Tx '" lim I c <X,f I>g
m-+<» n=l n n n
-At can be represented by
00 -:\ t I e k (u,e )e
n=l n n
and is obviously nuclear.
n
in 82
•
: Bl -+ 82 is
a sequence
that
-A t k
e
c) Our aim is now to construct an invertible nuclear operator veAl of
the form
00
v (A) u
-1 in such a way that v(A) E B+(A).
Define integers N(n), n 0,1,2, ••• , such that N(O);:: 0, N(n + 1) > N(n),
A , ) 1 > A ) and N(n + N(n
00 - A . lin L e J
j =N (n)
1 < '2 .
n
< 00
- 25 -
Now define the sequence {Vk
} by
Ak n
N (n - 1) < k S; N (n) , n = 0,1,2, ••••
00
It is clear that E vk
is convergent. k=l
Define a step function cr € B by setting cr(A) = en +
Ak_ 1 + Ak Ak + Ak+1
on A1
and cr(A) = e (2 2 3, k = 2,3, ••• -1
It is easily seen that cr(A) = veAl.
the interval A + A
on [0, 1 2 J 2
d) S is a nuclear space iff for every continuous semi-norm. p on S there
is another semi-norm q z p such that the canonical injection S ~ S q r'
is a nuclear map. Here the Banach space S is defined as the completion p
of the quotient space S;{p-l(O)}. Since the composition of a nuclear
operator with a bounded operator is again nuclear, $ is a nuclear space
iff for every semi-norm P1J!' 1jJ € B +' there
such that the canonical injection J : S
This canonical can be Px
injection written
(I':) Ju 00
I X-~(A.)1J!~(Aj) (x(A)u, j=l J
{x-~(A>e.} ~ J
We remind that
S respectively Px
Hilbert spaces
becomes
00
is a semi-norm p , X E.B+,X X
~ S is nuclear. P1jJ.
are orthonormal bases in the 2
we take X = a 1J! then (*)
\' -1 -~-~ Ju I... cr (A.)(x(A)u,x (A)e.)o1J! p .. )e.
1 ] J. J J
-1 And this operator is nuclear because cr (A
j) = V
j•
?
~) Suppose S is nuclear. See the definition in d} above. Take for p the X-norm. -Then for some semi-norm q the injection S ~ X is nuclear map. Further we q
can take 1jJ f. B such that p,/, ? q. (Apply Theorem 1.4.II). The injection + 'I' -1
S ~ X must be mlclear. Therefore tjJ (A) must be a nuclear operator and Pw -At -1 -At
consequently also H.S. But then: e = 1J!(A) [1J!(A)e ] must be H.S.
since the operator in square brackets is bounded for every t > O.
1jJ
Il
- 26 -
CHAPTER 2. The space TX,A
We now introduce our space of trajectories TX,A" The elements in this
space are candidates for becoming 'generalized functions" (or "distributions").
DeUni tion 2. I •
TX A denotes the complex vector space which consists of all mappings , +
F : ~ + X such that
(1) F is holomorphic.
(ii) + Vt,T E: a: M F(T) t
F (t + T) •
Such a mapping F will be called a trajectory_ If for some ~ E a:+ FI~) = F2(~) + then FI = F2 throughout a: • This follows immediately from the semi-group
and analyticity properties. We shall use the notation MtF for F(t}" It is
immediate that Vt > 0 MtF E S. In chapters 2 and 3 TX,A is abbreviated
by T.
Definition 2.2.
The embedding emb + and t t- ~
X + T is defined by (emb x) (t) = Mtx for all x E X
Sometimes we shall omit the symbol emb and loosely consider X as a subset
of T. Thus SeX c T. The question arises whether there exist trajectories F E T such that F(t)
does not converge or, even II F (t) II t 00 if t -Ii O. The answer is affirmative:
simply take x E X\DCA) and F(t} = Mtx. More general examples are obtained + by taking ~ B, u E X and defining G(t) = ~(A)Mtu, t E ~ • The next
theorem shows that all elements of T arise in this way.
Theorem 2.3.
For every F E T there exists w E X and $ E B+ such that F(t) + for every t E ~ •
Proof
-At = ljJ(A) e w
If the integral f~ 2A A e d(E
AF(1) ,F(1» converges then F(1) E D(e ).
- 27 -
A Take w = e F(l) and ~ = 1. Now suppose this integral is divergent. Take
00 2A - 2Tn a sequence {Tn}' Tn ~ O. All the integrals foe e d(EAF(l),F(l»,
n = 1,2, ••• , converge. Define a sequence of real numbers {N(n)},
n c 0,1,2, ••• , N(n) t 00 such that N(O) = 0, N(n) > N(n - 1) and
00
f 2" -2T 1A n+
e e d(EAF(1),F(1»
1 < 2'"
n N(n)
TIA Define IjJ on [0,"") by e
+ n = 1,2,... • Then IjJ E B
1-'(1) Eo: D(1/I-l (A)e~. Take
Tn+1A on [O,N(l)J and e on (N(n),N(n + 1)J,
and f~ e2A
IjJ-2(A)d(E"F(1),F(1» < 00 , which means
-1 A w = IjJ (A)e"F(1) •
In T we introduce the topology of uniform convergence on compacta in ~+. Because of the semi-group property F + F in Tiff vm E ~ F (l) + F(l) n n m m in the X-norm. More precisely
Definition 2.4.
n
The strong topology in T is the topology induced by the semi-norms Pn' n Eo: m I
In other words a base of open O-neighbourhood is given by
{FIIIF(';')11 < c., c > 0, 1 ~ j ~ N , ] ] j
Theorem 2.5.
I. T endowed with the strong topology is a Frechet space
II. A base of open sets {OV,t} is given by 0V,t = {FIF(t) E V, V open in X, t > 0 and fixed}.
(In words: the set of trajections which pass at t through V).
Each open set in T is a denumerable union of sets of this type.
- 28 -
Proof
I. Frechet means metrizable and complete. We define the distance of F from
the origin in the standard way by
00
d (F) = L 2 -~I F (l) 1I{1 + II F (l) II } -1 • n=l n n
The set of F with d(F) <
W • On the other c
1c
2",c
N
2-N- 1min(l,c1
, ••• ,CN
) certainly lies within
hand each W lies within a ball centered c 1c 2 ···cN
at the origin with radius
N 1 c 1 I n +-2
n 1 + cn 2N n=l
Therefore the sets n = {F\d(F) < 1 n € IN} establish a base of O-neighbour-n n
hoods. Further, since T is metrizable it is complete iff each Cauchy sequence
converges. Let {F } be m
Then for each n IN
a Cauchy sequence in T, 1 {F (-)} is a Cauchy sequence in X which converges to
m n an element h
1/ n E X. For V > none
sides converge and e«1/n)-(1/V»A
has F (l) = e«l/n)-(l/V»AF (1)· Since both m V m n
is a closed operator we have
h - D( «1/n)-{1/v»A) d h lin E e an 1/v = e«1/n)-(1/V»~1/n' Now we define
F(t) = e-(t-(1/V»~1/V'
Clearly F E T and is the limit of {F }. m
II. We proceed in three steps.
Re t € [1, 1 1) , V E IN • v v-
a} First we prove that, given 0V,t' then for each T, 0 < T < t, there
exists an open set V c X such that ° = a • For V we take the V,tA(t V,. A(t )
inverse image of V under the mapping e- -TJ, then e- -T (V) c V.
It is clear that each trajectory which passes at t through V, passes
at T through V and conversely.
b) We prove that aU,l/n' U c X open, n € IN, is an open set: V can be
covered by open balls lying inside V, therefore aV,l/n is a union of
translates of sets of t'l'pe W • cl,··eN
- 29 -
c) From a) and b) we,conclude that the sets aU t are open. Further , a n a with 1: < t can U,t V,T be written as ~U n °v = a~U V .
,1' f: n ,1:
Here U is the inverse image of U under e-(t-1')A. Now it is easy
to see that each translate of a set of type W is a set of c ••• c
N type aU,t' We are ready if we show that an arbItrary union of
sets aU,t can be reduced to a denumerable union of sets of this
type. This can be achieved by the construction
U (j
1 1 U,t n + 1 !S t<n
(j~1 ..L 'n+l
with
v U U 1 1
< t<-n + 1 n
1 1.) ..... where for each t £ [-;;r , U denotes the reduction of U at n
1 the point t to the point - in the way we described in a) • n+l "
Theorem 2.6.
emb(S) is everywhere dense in T.
Proof
For any F E T the sequence emb(F(l/n» converges to F in the strong
topology.
Theorem 2.7.
A set BeT is buunded iff for every t > 0 the set {F(t) IF E B} is
bounded in X.
Proof
~) Each continuous semi-norm has to be bounded on B.
Therefore Pn (F) ;: II F (lin) II is a bounded function on B.
r
Ll
- 30 -
Because of the boundedness of M for each T > a it then follows 't
that {F(t) IF E B} is a bounded set for each fixed t > D.
-) Suppose for each fixed t > 0 the set {II F (t)1I IF E B} is bounded.
We have to prove that each open a-neighbourhood can be blowed up
that B is contained 1n it. Let at = sup II F(t)1i 1 t fixed. FEB
00 -n within an open ball with radius E n=l 2 Cl 1/ n (1 + Cl 1/ n )
Example
Let B be a bounded set in X. Let ~ E B. Then the set
{FIF(t) = ~(A)Mtx , x € a} is bounded in T.
Theorem 2.8.
-1
Then B
A set K c T is compact iff for each t > 0 the set {F(t) IF € K} is
compact in X.
Pl:oof. -~) If K is compact then each sequence {F } c K has a convergent sub-
n sequence. This means that in the set K
t = {F(t) IF E K, t fixed}
each sequence has a convergent subsequence, which says that Kt is
compact in X •
such
lies
0
• ) Let {F } be a sequence in K. We must prove the existence of a converging n
subsequence. Consider the sequence {Fn (1)} c Kl c X. Kl is compact
therefore a convergent subsequence in Kl exists. Denote it by {F~(l)}. The sequence {F~(~)} has a convergent subsequence in K~o Denote it
bY{F2(~)}. n
Proceeding in this way we m t {F } c {F } for m > t and n n
arrive at sequences m 1 } {F (-) converges to n m
The "diagonal sequen ce II {1<,n} has the property n
{Fm} c K such that n
an element in Kl/mo
that {Fn(t)} converges n
to F(t) E Kt • But then also F~ + F in the strong topology.
I.
II.
III.
- 31 -
Example
Let K be a compact set in X. Let I/J E B. Then the set {FIF(t) == I/J(A)Mtx, x Kj
is compact in T.
In the last theorem of this chapter we describe the properties of our
topological vectorspace T in the standard termlnologyoftopological
vectorspaces. [SCH].
Theorem 2.9.
TX,A is bornological
TX,A is barreled
TX,A is Montel iff for t > 0 the operator Mt
-At is every = e
compact as an operator on X.
IV. TX,A is Nuclear iff for every t > 0 the operator Mt a H.S.-operator on X.
-At ;: e is
Proof
I,ll. T is bornological and barreled because it is metrizable. For
a simple proof see [SCH] 11.8.
III ~) Suppose T is Montel. Take a bounded sequence {x } c X. The n
sequence {Fn} defined by Fn(t) == Mtxn
is a bounded set in T. Because
of our assumption the closure of this set is compact. But then, by
theorem 2.8, for each t > 0 the sequence F (t) contains a converging n
subsequence. This shows the compactness of Mt
•
-) Suppose Mt is compact for every t > O.
Let B be a closed and bounded set. Then the set
Bt
= {F(t) IF E B, t > 0 and fixed} is bounded. Since BtH - M .. /Bt >
it follows that B is precompect in X. Now take any sequence {G } c B. t+T n
Because of the precompactness of each Bt
the "diagonal procedure"
of the proof of theorem 2.8 yields a converging subsequence of {G }. n
This subsequence converges to an element in B because B is closed.
We conclude that B is compact.
- 32 -
IV =» Suppose T is a nuclear space. Take n E :IN. There exists a
semi-norm P ~ P such that the natural injection T ~ T . = Xisn p p
nuclear. However there exists m > n and a > 0 such thattRe semi-
norm aPm~ p. Since the composition of a nuclear and a bounded
map is nuclear the natural injection t ~ f is nuclear.
This natural" injection is realized by Pm the Pn
map M(l/n)-(l/m) which
must therefore be nuclear. It follows that Mt
is H.S. for each t > O.
is
of the semi-group property
injection t ~ TPm Pn
is therefore nuclear. Now let p be anrealized by M(l/n)-(l/m) and
arbitrary semi-norm on T.There e?Cist a > 0 and n E :IN such that ap > P on T. Then for In > nn_ _we have ap > p and the natural injection T ~ T is nuclear. 0
m aPinP
• ) Suppose for each t > 0 Mt
is H.S. Because
Mt
is also nuclear. Let m > n. The natural
I
- 33 -
CHAPTER 3. The pairing of SX,A and TX,A
We now consider a pairing of Sand T. It turns out that S and T can
be considered as the strong topological dual spaces of each other.
Defini tion 3.1.
On SX,A x TX,A we introduce a sesquilinear form by
Note that this definition makes sense for t sufficiently small. Note
also that because of the semi-group property and self-adjointness of
Mt the definition does not depend on the choice of t. We remark that
<UO,F> = 0 for all F E T implies U o = 0 and <U,FO
> = 0 for all u E S
implies FO = O. These two facts easily follow by taking F = emb(uO)'
respectively the denseness of emb(S) in T.
Theorem 3.2.
I. For each F E T the linear functional <u,F>, which maps S onto 0;, is
continuous in the strong topology of S.
II. For each strongly continuous linear functional t on S there exists
GET such that t(u) = <u,G> for all u E S.
III. For each v E S the linear functional <V,G> , which maps Tonto 0;, is
continuous in the strong topology of T.
IV. For each strongly continuous linear functional m on T there exists
w E S such that m(F) = <v,F> for all F E T.
Proof
I. <u,F> is continuous on S iff for all t > 0 it is continuous in the
Xt-norm when restricted to this subspace. Indeed
- 34 -
II. For fixed t > 0 the mapping Mt
: X ~ S is continuous. Therefore the
mapping t(Mtx) : X ~ ~ is continuous. From Riess' theorem follows
the existence of bt E X such that l(Mtx) = (x,b t) for all x E X.
If we replace x by MtYr y E: X, it follows from the self-adjointness
of M that b == M b We take G such that G (t) = bt
, Then 1: t+T l t'
R..(u) = <u,G> for all u E S.
III, Since T is metrizable it is sufficient to prove continuity for
sequences in T, LetG ~ ___ n
small t we have <v,G > n
o in the strong topology. Then for sufficiently
(M tV,G (t» ~ 0 because G (t) ~ 0 in X-sense. - n n
+ . IV. Take WEB. Take a sequence {x } c X, x ~ 0 in X. Define w(A)x E T
n n n by (w(A)x ) (t) == w(A)M x • Since w(A)M
t is a bounded operator w(A)x ~ 0
n t n n strongly in T. By taking W = 1 we conclude first that the restriction
of m to X has the Riess representation m(x) = (x,S) for some a E X.
Secondly we conclude that m(w(A)x) is a continuous function on X for
each W E B+. Using the self-adjointness weAl it follows that a E D(w(A)l + for each W E: B • But then, by Lemma 1.10, e E S. Finally, as X is
denae in T the representation m(F) = (F,a) = <e,F> is valid for all
F t. T.
Definition 3.3.
The weak topology on S is the topology induced by the semi-norms
PF{u) = I<U,F>I, F E T. The weak topology on T is the topology induced by the semi-norms
p {G} = !<v,G>!, v E S. v
A simple standard argument, [CH] II § 22, shows that the weakly
continuous functionals on S are all obtained by pairing with elements
of T and vice versa. Together with Theorem 3.2 it follows then that
Sand T are reflexive both in the strong and the weak topology.
T S I. In the sequel we shall denote by
o
In the next theorem we show that in both spaces Sand SI weakly bounded
sets are strongly bounded.
- 35 -
Theorem 3.4. (Banach-Steinhaus)
I. Let :.: c $' be such that for each g EO S there exists M > o such that g for every F ( ::: one has I <g ,F> I :£ M , then for each t > 0 there exists g C
t > 0 such that for every FE::: one has II F (t) II :£ Ct
II. Let 8 c S be such that for every F EO SI there exists MF > 0 such that
for every f EO 8 one has ! <f ,F>!:£ MF
, then there exists T > 0 and c > 0 -such that 8 c X and II f II < c for all f E 8.
T T
Proof
I. From the assumption it follows that for each hEX and each t > 0 there
exists a constant Mh > 0 such that for every FE::: !<M h,F>! ~ ~ t ,t t n, or I (h,F(t»! :::; ~ • From the Banach-Steinhaus theorem in Hilbert h,t space it then follows that the set {F(t) IF E :::}, t fixed, is a bounded
set in X.
+ 2 2 2 II. Let t/i E: B • Let w EO D(t/i(A». Then t/i(A) w, defined by (t/i(A) w) (t) =l)!(A) I\W,
belongs to 5'. From our assumption it follows that for every
fEe l<f,l)!(A)2w>1 = I (l)!(A)f,l)!(A)w) I < M". • 'I' ,w
From the Banach-steinhaus theorem in Hilbert space it then follows that
e is a bounded set in the Hilbert space \p' i. e. the completion of S with
respect to the norm 1Il)!(A)· II. But this means that 8 is bounded in S,
since each semi-norm Plj!' + .
l)! E B ,is bounded on it.
In the next two theorems we give characterizations of weak convergence
of sequences both in S and S'.
Theorem 3.5.
Proof
U ~ 0 in the weak topology of S iff n
3t > 0 {U } c X and Vw EO Xt n t
Weak convergence of {Un} in Xt means weak convergence of M_tun in X.
- 36 -
~) Let F E S I. <u ,F> = (M u ,F(t». Since M tU + 0 weakly in X n -t n - n
and F(t) £ X, it follows that <u ,F> + O. n
.) First we remark that weak convergence in S implies
in X. Therefore for any w E X any L > 0, t > 0 fOL
as n .~ 00 {u} is a bounded set in S. Therefore n
weak convergence At
e d(E>..un,w) + 0
3T > 0 38 > 0 Vn E :IN II u II ~ e. We shall prove that n T
u + 0 weakly n
in XT
• Denote the projection operator J~ dE>.. by TIL' TIL commutes with
M • Take w £ X then -T
L
(M u ,w) 0 -T n I e>..td(E>..un,w) + (TILM_run,TILw)o·
o
We have II fiLM u II:;; 8 and II ITLw II + 0 as L + 0). Therefore if we take -r n L large enough the second term in (*) is smaller than ~E uniformly
for all n. As we have just seen the first term in C*) can be made
smaller than ~E by taking n large enough. This finishes the proof. 0
Corollary 3.6.
I. Strong convergence of a sequence in S implies its weak convergence.
II. Any bounded sequence in S has a weakly converging subsequence.
Theorem 3.7.
Proof
F + 0 in the weak topology of S' iff n
Vt > 0 Vv E X
.) For any v E X <M v,F >= (v,F (t»O + 0 as n + 00. t n n
.. ) For any <P E Sand t sufficient.ly small
because M -t qJ X.
- 37 -
Corollary 3.8.
I. Strong convergence of a sequence in $' implies its weak convergence.
II. Any bounded sequence in S' has a weakly convergent subsequence. (Use
a diagonal argument).
It looks reasonable to conjecture that the weak topology on S is the
inductive limit topology with respect to the Hilbert spaces Xt , now
endowed with the weak topology. It is easily seen that the weak
topology on 5' is induced by the semi-norms Pt
' t > 0, V € X ,v and Pt,v(F) = I {v,F{t})ol. We will not pursue these things further
here. The next theorem deals with the question: When does weak convergence
of a sequence imply its strong convergence?
Theorem 3.9.
The following three statements are equivalent.
I. For each t > 0 -At
e is a compact operator on X.
II. Each weakly convergent sequence in S converges strongly in S.
III. Each weakly convergent sequence in $' converges strongly in S'.
Proof
I - II. A weakly convergent sequence in S converges, for some t > 0,
weakly in Xt • See Theorem 3.5. Because of the assumption the natural
injection Xt
c Xa
, 0 < a < t, is compact. Cf. the proof of Theorem 1.11.
But then our sequence converges strongly in X • a
II ~ I. Take any sequence {f } C n
t > 0 and each F E S' we
X, f -+ 0 weakly in X. For each n
have <M f ,F> -+ O. Thus Mtf -+ 0 weakly in 5. t n n
Because of the assumption Mtfn -+ 0 strongly in S. And so IIMtf
n '6 -+ O.
This shows that Mt must be compact.
I ~ III. Let {F } C S'. Suppose Vg € S <g,F > -+ O. n n
Then Vh E X Va > 0 <M h,F > (h,F (a» -+ O. This means that Va > 0 ann
Fn(a) + 0 weakly in X. Using the compactness of MS' S > 0, we find that
F (a + B) = MoF (a) -+ 0 strongly in X. Therefore Vt > G IIF (t) \I -+ O. n ~n n
- 38 -
III ~ I. Let {v } c X be such that v n n 7 0 weakly in X. Then v 7 0
n weakly in 5' because Yg E S <g,v> = (g,V )0 7 O. Now the n n assumption says vn 7 0 strongly in S'. This means Vt > 0 Mtv
n 7 0
strongly in X. The compactness of Mt follows for each t > O.
One might wonder whether in the case that Mt
is compact for each
t > 0 the strong and weak topologies coincide. This however cannot
be true since a set which belongs to a weak base of O-neighbourhoods
always contains a closed subspace of finite codimension • Generally
speaking such "large" sets do not fit in a strongly open O-neighbour
hood.
o
It is relatively simple to show that both in Sand $1 the weakly compact
sets are just the (weakly) closed and bounded sets. Then it is not
difficult to prove that Sand S' with their strong topologies are
Mackey spaces. That is the strong topology is the finest locally
convex topology for which the dual of S (resp. 51) is 51 (resp. S). (All spaces which are either barreled or bornological are Mackey.
See [SCH] IV.4).
- 39 -
CHAPTER 4. Characterization of continuous linear maRpings between the soaces
Let B be a non-negative self-adjoint operator in a separable Hilbert
space Y. As before A is a non-negative self-adjoint operator in a
separable Hilbert space X. In this chapter we shall derive conditions
implying the continuity of linear mappings SX,A + Sy,B' SX,A + Sy,B ' S~,A + SY,B' S~,A + Sy,B' Further we investigate which linear
operators, defined on a subset of X, can be continuously extended to
operators on S~,A' The next theorem is an immediate consequence of the
fact that SX,A is bornological. Cf. Theorem 1.11, For completeness we
give an ad hoc proof.
Theorem 4.1.
A linear map Q : SX,A + V, where V is an arbitrary locally convex
topological vector space, is continuous
1. iff for each t > 0 the map Q!\ : X + V is continuous.
II. iff for each null sequence {un} c SX,A ' un + 0 in SX,A ' the sequence
{Qu } is a null sequence in V. n
Proof
1 .. ) Mt is an isomorphism from X to Xt
' By the definition of the
inductive limit, Xt
is continuously injected in SX,A' SO if Q is
continuous it follows that QMt is continuous •
• ) Let ~ denote the restriction of Q to Xt
• From the continuity of
QfAt on X follows the continuity of Qt on Xt ' Now let rt 3 0 be open in -1 -1 V. For each t > 0 Q Crt) n X
t = ~ (n) is an open a-neighbourhood in
-1 Xt ' Thus Q (rt) is open in SX,A'
II) Follows from I because null sequences in S are always null
sequences in some Xt
' t > 0 and vice versa.
In the next theorem we characterize continuous linear mappings from -Bt SX,A to SY,B' We denote Nt = e ,
r' l'
- 40 -
'l'heorem 4. 2 ,
Suppose P : SX,A ~ Sy,B is a linear mapping. The following six conditions
are equivalent.
I. P 1s continuous with respect to the strong topologies of SX,A and SY,B'
II. un ~ 0, strongly in SX,A' implies Pun ~ 0, strongly in SY,B'
III, For each a > 0 the operator PM is a bounded linear operator from a
X into Y.
IV. For each t > 0 there exists B > 0 such that PMtCX) c Ne{Y) and N_ePMt is a bounded linear operator from X into Y.
V. There exists a dense linear subspace E c V such that for each fixed
y € - the linear functional Lp,y(f) = (Pf,y)V is continuous on SX,A"
* VI. For each t > 0 (PMt ) is a bounded linear operator from Y to X. * (Remark: One has (PMt ) (V) c SX,A).
Proof
We proceed according to the scheme
.............,. ~ -;::::~~ ~ I II III IV V VI ~~
I ~ II. See Theorem 4.1.
II .. III. If {x } is a null sequence in X then for any a > 0 {M x } is nan a null sequence in Sx A' By II {PM x } is a null sequence in , a n SY,B and hence in Y.
III .. IV. Let a > O. We define a class A of continuous linear a
functionals on X in the following way
Obviously a 1 < a 2 .. Aa1 c Aa2 since for g € Na1 CV) we have
N q N (N g) where N g € N (V) and 1\ N g lIy :$ 1. We -a1 -a2 u2-a1 a 2-a1 a 2 a 2-a1
claim
Vx € X 3a > 0 3M > 0 V£€ A a I R, (x) I :$ M •
III - V.
- 41 -
Indeed, since PMtx E Sy,B' for et sufficiently small PMtx € Vet and
N PM x € Y so that -0. t
1R.(x) I = I (N PM x,g)vi $ liN PM xliV -0. t -0. t
Applying a modified version of the Banach-Steinhaus theorem, see [J],
see our Appendix C, from (*) it follows
30. > 0 3M > 0 Vx € X V~ € A 0.
But then for all x E X and all g € Y 0.
which means PMtx to D(N ) and N -clMt -0. We can 1!ake - :: Y. Take y € Y fixed.
t > 0 we have
I Lp (M x) I = I (PMtx ,y) V I s ,y t
I R. (x) I ~ MIl x \I
bounded.
For each x € X and each
Ct,yll xliX
Together with Theorem 4.1 the result follows.
V - VI. According to Riess' theorem for each y € 3 and each t > 0 there
exists f t € X such that for each h € X
Replacing h by M x, X € X we observe that f = M f t so that f t E Sx A' T * t+T T . ,
From (*) we obtain ft
= (PM) y. SO D«PM )*) ~ E which is dense in V. t t *
Since PMt
is defined on the whole X the operator (PMt
) is defined
on the whole V and bounded. Repeating the argument with arbitrary
* y € X shows (PMt ) y E SX,A' VI - III. PMt is bounded because (PMt )* is bounded.
IV - II. Let {un} be a nUll-sequence in SX,A" Then for some 0. > 0
{M u} is a nUll-sequence in X. But then, using the boundedness of -0. n N_SPMo.' {N_SPun } = {N_SPMo.M_o.un} is a null-sequence in V. Hence {PUn}
S [1 is a nUll-sequence in Y,So-
I.
II.
III.
- 42 -
The next corollary is important for applications.
Corollary 4.3.
Suppose Q is a densely defined closable operator: X ~ Y. If
D(Q) J SX,A and Q(SX,A) c SY,B' then Q maps SX,A continuously into
SY,B'
Proof
Q is closable iff D(Q*) is dense in V. Since QMt : X ~ Y is defined
* on the whole X, it.s adjoint (!2.Mt) is bounded. T*he adjoint however
* * is densely defined since on D(Q*) one has (aMt ) = MtQ • Hence (~\{t)
is defined on the whole Y and bounded. From this the boundedness of
!2.Mt
follows. Application of Theorem 4.2 III yields the desired result. n
Theorem 4.4.
Let K : SX,A + Sy,B be a linear mapping. The following four conditions are
equivalent.
K is continuous with respect to the strong topologies of SX,A and SY,B'
For each t > 0, a > a NtKMa is a bounded linear operator from X into Y.
For each t > a NtK is a continuous map from SX,A into Sy B' , IV. There exists a mapping Z of the interval (o,~) into the continuous linear
maps from SX,A into SY,B such that
(i) For each t > 0, T > a Z(t + t) = NtZ(T) •
(ii) For each u E SX,A Ku = lim emb(Z(t)u) t-l-O
strongly in Sy B' ,
Proof
I - II. Let {x } be a null-sequence in X. Then {M x } is a null sequence nan in SX,A' Since K is continuous {KMaxn} is a null sequence in Sy,B' which means that for each t > 0 {(N KM )x } is a null sequence in Y.
tan Hence N KM is bounded.
t a
- 43 -
II ~ I. Let {Un} be a null sequence in SX,Ao Then for some a > 0
{M u } 1s defined and is a nUll-sequence in Xo But then -et n {Kun } = {KMetM_aun } is a null-sequence in Sy,B since for each t > 0
(KM M u) (t) = (NtKM)M u ~ 0 in Y et -a n a -a n
II ~ III. Apply theorem 4.2 with P = NtKo III ~ IV. Define Z(t) = NtK. It obviously has the property (i). we
must show that for each, > 0 lim N Z(t)u = (Ku) (,). But this is also HO -r
obvious because N~Z(t)u = Nt Ku and the semi-group N is strongly ,+, u continuous.
IV ~ III. Trivial because NtK Z (t) •
Theorem 4.5.
Let r : SX,A ~ Sy,B be a linear mapping.
Let r r
X ~ Y denote the restriction of r to X. The following five
conditions are equivalent.
I. r is continuous with respect to the strong topologies of SX,A and
SY,B"
II. r;(Y) c SX,Ao
III. There exists t > 0 such that r;(Y) c Mt(X) and M_tr; is a bounded
operator from Y into X.
IV. There exists t > 0 such that r M r -t
with domain Xt
c X is bounded as
an operator from X into Y.
V. 'l'here exists t > 0 and a continuous linear map Q SX,A ~ Sy B such
that r = w\.
Proof
I - II. From the continuity of r it * continuous. Its adjoint f , r
immediately follows that r is r
,
* defined by (x,rrY)X = (rrx,y)y for all
x E X, Y E Y, is a bounded operator as well. Now consider the dual
operator r' : SY,B ~ SX,A defined by
<F,r'G>X <fF,G>y for all F E SX,A'
o
- 44 -
For fixed G E Sly,B the righthand-side defines a continuous linear
functional on SX,A' r; is a restriction of r 1 and therefore maps
Y into the set SX,A C X. II ~ III. For e > a consider the following class of continuous linear functionals
on y
As in the proof of Theorem 4.2 one shows that
We claim
Vy E Y 313 > a I R, (y) I ~ M •
* Indeed, since rrY E SX,A one has for 13 sufficiently small
Applying the modified Banach-Steinhaus theorem, see Appendix C, it
follows
3S > a 3M > 0 Vy E Y V2 E Be I R, (y) I ~ M Uyll y •
But then for all f E MS(X) and all y E Y
which implies r;y E D(M_ S) = Xs and M_er; is bounded.
* * * * * III - IV. M_arr has a bounded adjoint (M Qr) but (M Qr) ~ r M Q ~ -~ r -~ r r -~
with domain Xe C x. * * IV - V. Take Q = (M or ) . For each a > 0 ~M is the extension of
-~ r ~ a r M Q which is bounded, So Q satisfies condition III of Theorem 4.2.
r -Io'+a We have r = Q.M~.
V - I. If r = Q.Mt the mapping r is obviously continuous. n
- 45 -
Theorem 4.6.
Let 4l: S X,A -+ SY,B be a linear mapping.
Let ~r X -+ Sy,B denote the restriction of 4l to X.
The following six conditions are equivalent.
1. 4l is continuous with respect to the strong topologies of
SX,A' SX,A and
II. For each g E Sy,B the expression <g/~F>yt F € SX,A is a continuous
linear functional on SX,A
III. For each t > 0 the operator Nt 4l is a continuous map from SX,A into
SY,B' *
C SXtA • IV. For each t > 0 (Nt
4lr
) (y)
* c Me (X) V. For each t > 0 there exists e > a such that (Nt 4lr ) (y)
M-/3(Nt~r) * operator from Y into X. and is a bounded
VI. For each t > 0 there exists 13 > 0 such that N 4l M 0 = N ~M Q on t r -p t-p
the domain X/3 C X is bounded as an operator from X into Y.
Proof.
We proceed according to the scheme
~~-....~ I II III IV V VI ~
I ~ II. Trivial.
I ~ III. Trivial because Nt : Sy,B -+ Sy,B is continuous.
III ~ IV. Theorem 5.4. condition II.
IV ~ V. Apply Theorem 4.5 to N ~ . t r
* V - VI. The adjoint of the bounded operator M-13(Nt~r) is an extension of (Nt
4lr
}M_S
'
Therefore the latter is bounded.
VI ~ I. Let {Fn} be a null sequence in SX,A' Then for each t > 0
there exists S > 0 such that we can write (~F ) (t) = Nt~ M of (13). n r -p n This converges to zero as n -+ ~ because of the boundedness of Nt 4lr M_S'
II ~ V.
- 46 -
<q,~F>y has the representation <f,F>X' see Theorem 3.2.IV,
here f = ~Ig E SX,A' Taking F = U E SX,A we observe that
(u,~'g)X = <u,~'g>X = <~u,g>y is, as a function of g, a continuous
linear fUnctional on Sy,B' Then with Theorem 4.2.V it follows that
~' maps SV,B continuously into SX,A" Then, by Theorem 4.2.IV, for
each t > 0 there exists S > 0 such that M_B~'Nt is a bounded
* operator. But M_e~INt = M_B(Nt~r) because for all x E X, Y € V
* (~INtY'x)X = <NtY,$rx>V = (YINt~rx)y = «Nt~r) y,x)X
The interesting question arises which densely defined (possibly
unbounded) operators from X into Y can be extended to a continuous
mapping from SX,A into SY,B'
Theorem 4.7.
Let E be a linear map X ~ D(E)
to a continuous linear map t : + Y with D1tf = X. E can be extended
SX,A + Sy,B iff E has a densely
D{E*, + X with E*(Sy,B) C SX,Ao E* .' Y S defined adjoint ~ V,B ~
Proof
-) For each ~ E D(E) and g € Sy Bone has (E~,g)y = (~,E*g)X.· , *
It follows from Theorem 4.2.V that E maps Sy,B continuously into
SX,A' We define EF, F € $'
X,A by
<v,IF>y * <E V,F>X
This functional is continuous as a function of v, so IF € Sy B' , It is obviously continuous as a function of F. Then by Theorem 4.6.11
the continuity of I follows.
-) If E exists as a continuous map, its dual operator E' maps Sy,B
into SX,A'
For each x € D(E) and g E Sy,B one has <g,Ex>y = (g,Ex)V * -It follows that E ~ E' and E*(SY,B) c SX,A'
n
- 47 -
Corollary 4,8,
A continuous linear map Q : SX,A + SY,B can be extended to a
continuous linear map Q: Sx A + Sy B iff Q has a Hilbert space
* * ' *' adjoint Q with D(Q ) ~ SY,B and Q (SY,B) c SX,A'
- 48 -
CHAPTER 5. Topological tensor products of spaces of type 5' 5' . X"A'. X(A
For two separable Hilbert spaces X and V we consider the complex
vector space consisting of all Hilbert-Smidt operators Z from X into
V • vJe shall denote this vector space by X ® V. For any Z e: X ® V
and any orthonormal basis {ei
} C X we have
"" Olzlll2 = .I IIzeil~ < "".
~=1
The double norm III • III does not depend on the choice of the orthonormal
basis { e. }. We introduce an inner product in X e V by ~
00
(Z,K) XQV
Endowed with this inner product X ® V is a Hilbert space. See [RS]
cr.. VIII.1D. Examples of elements in X ® V are!; ® n, ~ e: X, n e: V
defined by (~en)f: (f'~)Xnf for all f E X, and finite linear N
combinations of these: l: j=l (F,; j ® ~) with f; j E X, nj E Y. The linear subspace
of X e Y which consists of all H.S. operators of the type just
mentioned will be denoted by X ®a V, i.e. the (sesquilinear)
algebraic tensor product of X and Y. X ® V may be regarded as the
completion of X ® V with respect to the double norm 111·111 • Therefore a
X®Y is called the completed (sesquiliriear.) topological tensor product
of X and Y. For later reference we mention the following properties
taken from [RS] Ch. VIII.
Properties 5.1.
(a)
(b)
Vx,.; £ X Vy,n E Y
VA E ~ VF,; E X Vn £ V
Thus the canonical mapping X x Y ~ X e V, defined by [x;y] ~ x e y
is anti-linear in x and linear in y.
- 49 -
( c) VZ E X ® Y Vx E X Vy E Y (Z,x ® y)X®Y = (ZX,Y)y
Let H resp. ] denote bounded linear operators on X, resp. Y into
themselves. H ® ] denotes the linear mapping of X ® Y into itself
defined by (H ® J) (x ® y) = (Hx) ® (Jy) and linear extension,
followed by continuous extension.
(d) The uniform operator norms of H, J and H ® J are related by
II H ® J II = II H1111J II •
(e) (H ® J)Z = JzH*
(f) Hand J injective -H ® J is injective.
The theory of closable tensor products of unbounded closable operators
and the description of their properties in terms of corresponding
properties of their factors presents greater difficulties. Only
rather recently significant general results have been attained [TJ.
Definition 5.2.
Let A with domain DCA) be a densely defined closed linear operator
X. Let B with domain D(B) be the same in Y. On D(A) ® D(B) c X ® Y a
we introduce the operator A ® r + r ® B by (A ® . r + 1 ® B) (x ® y) a a a a
(Ax) ® y + X ® (By) and linear extension. This extension is well
defined and closable, [RS] p. 298.
Lemma 5.3.
Let A resp. B be self-adjoint operators in X resp. Y.
I. A ® 1 + 1 ® B is essentially self-adjoint in X ® Y • We denote the a a
unique self-adjoint extension by A ® 1 + 1.:0 B or, briefly, A EEl B.
II. A ~ 0 and B ~ 0 implies A EEl B ~ 0
Proof
I. We follow Nelson's approach [SJ. The operator C = A ® r + I ® B a a
is obviously symmetric. C 1s essentially self-adjoint iff C*z = ± iZ
implies Z = O.
- 50 -
-iAt iBt On X ® Y we introduce the bounded linear operator e ® e
t E lR. This operator is isometric and
For Z E DCA) e D(B) we have a
* * Now let K E D(C ) and C K = ±iK
Then for each Z E D(A e r + I® B) a a
d -iAt iBt 'dt(Z, (e ® e )K) X®Y
-iAt iBt = =F (Z, (e ® e ) K) XeY
Thus
+t e (Z,K)XeY
But
1 -iAt iBt I
(Z I (e ® e ) K) X®y. $ II zll X®yll 1<11 X®Y
This contradicts (*) except when (Z,K)X®Y = 0 for all Z E D(A) ®a D(B),
which 1s a dense set. Therefore K must be zero.
II. We must show that sums of type
are non-negative. This follows by taking orthonormal bases in the
span of x1"",xN and Yl""'YN and straightforward calculation.
Since the graph of A e 1 + 1 ® B is dense in the graph of A EEl B a a
it follows that also the latter must be non-negative.
Theorem 5.4.
On X ® Y we have for t ~ 0
- (AEEl B)t -At -Bt e = e ® e = Mt ® Nt •
[1
- 51 -
Proof
-At -Bt e 0 e is a strongly continuous one-parameter semi-group of
bounded non-negative hermitean operators. Obviously
(e-At ® e-Bt ) (D(A) ® D(S» c D(A) ® D(S). The infinitesimal a a
generator of Mt ® Nt restricted to D(A) ®a D(B) is -C. The
infinitesimal generator must therefore contain the closure of -C
which is the self-adjoint operator ~A ~ B). The infinitesimal
generator however has to be self-adjoint. Hence it must be equal
to -(A ~ B). Mt ® Nt is, of course, holomorphic. 0
Applying the results of th~ preceding chapters we can introduce
the spaces SX®Y ,A~ H' SX®Y ,A IEB and, by taking A = 0 or B = 0,
the spaces SX®Y,A®I' SX®Y,I®B' etc.
Definition 5.5.
The canonical sesquilinear map ® : SX,A x Sy,B -+ SX®Y,A ~B is
defined by [UiV] -+ u ® v. Here the symbol ® is the same as in
Properties 5.1. This definition is consistent because for u E SX,A'
v E SY,B there exist x E X, Y E Y and t > 0 such that u = Mtx ,
v = Nty • Further u 0 v = (Mt
® Nt) (x ® y) = (Mtx) 0 (Nty). So
that u ® v E SX®Y,A ~B'
Theorem 5.6.
SX®Y,A 83B is a complete topological tensor product of SX,A and Sy,S'
By this we mean:
I. SX®Y,A EElB is complete.
II. The canonical sesquilinear map ®
continuous.
III. The span of the image of 0, i.e. the algebraic tensor product
SX,A ®a Sy,B ,is dense in SX®y,A~S'
- 52 -
Proof
I. The completeness follows from Theorem 1.11.
II. It is enough to check the continuity of S at [0;0]. Let W be a
convex open neighbourhood of 0 in SX®YIA~B.Then for each t > 0
W n (X ® Y)t is an open O-neighbourhood in (X ® Y)t and it contains
an open ball centered at 0 and radius r t , 0 < rt
< 1. In Xt
resp.
Yt we introduce open balls At resp. Bt' centered at 0 and with
radius both r t • Let A = U At C Sx A and B = U Bt C SY,B' t>O I t>O
Then ® maps A x B in W since II x ® y II i (t ) ~ "x lit" y II ~ r i (t ) _ _ m n ,'r 'r m n ,T
whenever x € A, y € B. Let A resp. B denote the convex hulls of A A "-
resp. B. Then ® maps A x B in W. The set A is convex and A n Xt
contains an open a-neighbourhood in Xt " From Theorem 1.4.11 it
follows that A contains an open set V". • - ~,£
Similarly B contains an open set V r' We conclude that ® maps X,u v x V into W.
1f;, €'. X, a III. For each t > 0 X
t ®a Y
t is dense in (X ® Y)t' From this the desired
result follows.
Remark
Our strong topology on SX®Y,A ~8 is, generally speaking,not the
projective tensor product topology. Cf. [SCH] p. 93. Therefore the
universal factorization property for continuous sesquilinear maps
on this space does not hold in general.
Definition 5.7.
The canonical sesquilinear map ® : SX,A x SY,8 -+ SX®Y,A83B '
[FIG] -+ F ® G, is defined by (F ® G) (t) = F(t) ® G(t) •
LJ
Here ® is the same as in properties 5.1. The definition is consistent
because (F ® G) (t + T)
=: (M ® N ) (F ® G) (t) • T T
= M F(t) ® N G(t) = (M ® N) (F(t) ® G(t»= 'r T T T
- 53 -
Theorem 5.8.
SX®Y,A£:I3B is a complete topological tensor product of SX,A and Sy,B'
By this we mean:
I. SX®Y,A!EB is complete.
II. The canonical sesquilinear map ®: SX,A x SY,B -+ SX®Y,A£:I3B is continuous.
III. The span of the image of ®, i.e. the algebraic tensor product
SX,A ® SY,B' is dense in SX~y,A £:I3B •
Proof
I. The completeness follows from Theorem 2.5.
II. For each t > 0 we have IIF(t) ~ G(t) I~&y == IF(t)llX"G{t)lIy' From this the
continuity at [O;oJ follows.
III. X @a Y is dense in X @ Y which is dense in SX@Y,A £:I3B'
We now want to introduce mixed sesquilinear topological tensor products
of type Sx A @ Sy B' etc. The reader who is already content with the , , Kernel Theorems case a and case b of the next chapter may skip the
remainder of this chapter.
Defini tion 5.9.
We introduce the following linear subspace of SX®Y,I&B
If no confusion is likely to arise we use the shorter notation ~B
o
for this space. For ~ E ES we have for each t,. > a ~(t + T) = (1 ® NT)~(t).
On 1B we introduce semi-norms {Pt,~}, t > 0, ~ E B+, by
and the corresponding locally convex topology.
- 54 -
Definition 5.10.
The canonical sesquilinear map ® : SX,A x Sy,B + ~B is defined by
($ ® G) (t) = $ ® G(t). Here the s~~ol ® is the same as in Properties
5.1. The span of the image of ® is denoted by SX,A ®a SY,B'
Theorem 5.11.
~B is a complete topological tensor product of SX,A and SYtB' By this
we maan
I. ~B 1s complete.
II. The canonical sesquilinear map ® SX,A x Sy,B + ~B is continuous.
III. SX,A ®a Sy,B is dense in ~B .
Proof
I. Let {~ } be a Cauchy net. The a's belong to a directed set D. First a
take 1/1 = 1. ~a tends to a limit point ~ E SX®Y,I®B because the latter
is complete. It remains to show that for each t > a ~ (t) E SX®Y,A EBB"
For each 1/1 E B and each t > 0 1/1 (A EEl B)~ converges in X ® Y. + a
From the closedness of 1/1 (A EEl B) it follows that ~(t) € D(1/I(A EEl B».
This is true for each 1/1 E B+ and therefore by Lemma 1.10, for each
t > 0, <p(t) E SX®Y,AEElB'
II. It is enough to check the continuity at [O;OJ. Let Wt ,I, be the set ''f'11::
of elements in ES for which the semi-norm Pt,1/I is smaller than 1::; it is
an open O-neighbourhood. Let W denote the convex open O-neighbourhood 1/I,e:
in SXGilY,A EBB consisting of elements with semi-norm P1/l smaller than 1::.
Then W consists of those trajectories in ~B' which pass at t t'!ete:
through W . According to the proof of part II of Theorem 5.6 there ljJ,e:
exist open O-neighbourhoods U c SX,A ' V c SY,B such that
U ~ V c W . V n YLt contains an open ball, centered at 0, of radius 0, ljJ I e: ""2
say. Now let n {GIG E SY,B' II G(~t)1I Y < o}. n is open in SY,B and
the set {G(t) \G E n} c V. It follows that <P ® G E Wt ,I, whenever $ E U I'f',e:
and GEn. This proves the continuity of ®.
- 55 -
III. SX,A ®a SY,B can be considered as a subspace of SX,A ®a Sy,B'
SX,A ®a Sy,B is a dense subspace of SX®Y,AEBB' See Theorem 5.6, So
we are ready if we show that SX~Y ,A EBB is densely embedded in ES' Let <P € ES then for t > 0 <l> (t) E SX~y,A EBB' We claim that 4> (T) -+ 4>
in LS-sense as , f O. Indeed
({I ® N - I 0 I}4>(t), ,
The first factor in this inner product tends to 0 as T + 0 because
1 ® Nt is a strongly continuous semi-group. The second factor remains
bounded as T + O. This is so because it can be written.
with t: sufficiently small. The operators between { } are (uniformly)
bounded. This finishes the proof,
Remark.
In the proofs of parts III of Theorems 5.6 and 5.11 we circumvented the
problem whether the strong topology on SX®Y,A EBB is induced by semi
norms of the form ~(A) ® X(B) with ~,X E B+,
Definition 5.12.
We introduce the following subspace of SX®Y,A®I
E' ={plp€S· Vt>O X 0 Y ,A01 ,A EBB X®Y ,A®I ,
o
If no confusion is likely to arise we use the shorter notation LA for
this space. For P E EA we have for each t,T >0 pet + T)"=: (M,0 I)P(t).
On EA we introduce the semi-norms Pt,~ of Definition 5.9 and the
corresponding locally convex topology.
- 56 -
Definition 5.13.
The canonical sesquilinear map ® : SX,A x Sy,B ~ LA is defined by
(F ® W) (t) = F(t) ® W. The span of the image of ® is denoted by
SX,A ®a SY,B'
The proof of the following theorem runs the same as the proof of Theorem 5. 11 •
Theorem 5.14 • . EA is a complete topological tensor product of SX,A and SY,B' By this
we mean
I. LA is complete.
II. The canonical sesquilinear map ® SX,A x Sy,B + LA is continuous.
III. SX,A ®a SY,B is dense in EA'
we conclude this chapter by introducing another candidate for a sesquilinear
tensor product of mixed type. For a > 0 the map M ® 1 which carries IX
SX®Y ,A EE1B into
SX®Y ,A EEl B into is realized by
itself can be extended to a continuous linear map from
itself. This follows from Corollary 4.8. The extension
«M ® I)K) (t) = (M ® I)K(t). The map M ® 1 is a IX a injective, see Properties 5.1.f, we denote its inverse by M ® 1.
-a
Definitions 5.15.
We introduce the locally convex topological vector space
The topology on (Ma ® 1) (SX®Y,A EEl B) is given by the metric d et • Here
det(T) = d«M_a ® 1)T) and d(') denotes the metric on SX®Y,AEElB' See
Theorem 2.5. The topology on EX®Y,A®I,AEBB or, briefly, EA is the
inductive limit topology with respect to the metries d , et > O. We will et
not look for an explicit system of semi-norms for LA' nor investigate
its completeness here.
- 57 -
On ~A x rA we introduce the sesquilinear form
for E > 0 sufficiently small. The definition does not depend on the
choice of E. Further we define the embedding of LA into LS by
(*) (emb T) (t) = (1 x Nt) T •
Theorem 5.16.
I. SX,A ®a Sy,S is dense in LA'
II. For each fixed P € LA the linear functional <P,T>A' as a function of T, is continuous on LA'
III. The embedding (*) of Definitions 5.15 is continuous.
Proof
I. X ® Y is dense in SX®Y A lEE' Therefore for a > 0 X ® Y is dense in a , a a
(Ma ® I) (SX®Y,A lEI B) " We conclude that U X ® Y which.1s a subset a>O a a
of SX,A ®a Sy B is dense in EA" t
II. It 1s sufficient to prove the continuity for restrictions to each
III.
(Ma ® I) (SX0 Y,A IES) , a > O.
We have \<P,T>A\ ~ cd(M_a 9 I)T) :::: cd (T) " a Here c is a constant which depends only on P and a •
P t ,II (emb T) = II {lidA lEI S) (M x 1) (1 0 Nt)} (M ® IYI'II for a sufficiently ''1' a -a X®y
small. The operator between { } is bounded. Therefore p (emb T) ~ C d (T) t,$. a
where c only depends on t, $ and a.
Definitions 5.17.
Similar to Definitions 5.15 we introduce the space
briefly dented by LS' with the appropriate inductive limit topology.
For [~;FJ € LS x LS we introduce the pairing ~,F>S = <~(£)/(1 x N_~)F> .. X®y
Further we define the embedding of IS into LA by (emb F) (t) = (Mt
@ I)F.
- 58 -
Analogous to Theorem 5.16 we have
Theorem 5.18. ~
I. SX,A e a SY,B is dense in LB'
II. For each fixed ~ E SX,A the linear functional <~,F>B' as a function
of F, is continuous on LB'
III. The embedding described in Def. 5.17 is continuous.
- 59 -
CHAPTER 6. Kerneltheorems
In this final chapter we show that the elements of the completed sesqui
linear topological tensor products of the preceding chapter can, in a
very natural way, be interpreted as linear maps of the types we discussed
in Chapter 4. We give necessary and sufficient conditions on the semi--tA -tB groups Mt = e and Nt = e which ensure that the topological tensor
products comprise all continuous linear maps. In this case we say that a
Kernel Theorem holds.
CASE a: Continuous linear maps SX,A + SY,B' We consider an element
a E SX®Y,A fEB as a linear operator SX,A + Sy,B in the following way,
Let F E SX,A' We define aF by
(a) aF = N (N aM ) F (e::) , e:: -E: -€:
For €: > 0 and sufficiently small this definition makes sense and does not
depend on e::,
Theorem 6.1 • . I. For each a E SX®Y,A fEB the linear operator a
(a), is continuous.
II. For each 8 E SXIli!IY,A fflB ' F E S I X,A ' G E SY,B
SX,A+ SY,B' as defined by
III. If for each t > 0 at least one of the operators Mt , Nt is H.S, then
SXIli!IY ,A fEB comprises all continuous linear operators from SX,A into
SY,B'
IV. SX0X,AfEA comprises all continuous linear operators from SX,A into SX,A
iff for each t > 0 the operator Mt is H.S.
Proof
I. We shall prove that a satisfies condition IV of Theorem 4.5, Since
- 60 -
8 X 0 Y we have 8 r 8. Since 8 € SX0Y,A IE B we have for t sufficiently
small N_tGM_ t € X ® Y. Therefore OM_ t = Nt(N_tOM_t ) is bounded.
II. For E sufficiently small N 8M € X «I Y, therefore by Properties -e -e
S.l.c.
<8F,G>y = < N (N 8M )F(£},G>y = E -e -e
III. Let r : Sx,A 7 Sy,B be continuous. By Theorem 4.S.V there exists T > 0
and continuous Q : Sx,A 7 Sy,B such that r = QMT' By Theorem 4.2.IV
there exists S > 0 such that N-eQM~T is a bounded operator. Put
~ = ~ min(S,~T), then
r = N {No (N SQML )ML }M ~ ~-~ - ~T ~T-~ ~
The operator between ( ) is bounded. Further the operator between { }
is B.S. since No or ML is H.S. ~-~ ~T-~
It follows that r E SX0Y,A IEB'
IV. The if-part is a special case of III. For the only if-part concider the
special map r = M~ Sx,A 7 SX,A for some ~ > O. In order that
r ( SX®X,A lEA it has to be H,S.
CASE b: Continuous linear maps SX,A 7 SY,B' Let K E SX®Y ,A IEB' For f E SX,A we define Kf € SY,B by
(b) (Kf) (t) = Nt_EK (e) M_Ef
For any f E SX,A and t > 0 this definition makes sense for E > 0
sufficiently small. Moreover (Kf) (t) does not depend on E.
Theorem 6.2.
I. For each K E SX0Y ,A Il3B the linear operator K
by (b) is continuous.
SX,A + SY,B defined
o
- 61 -
III. If for each t > 0 at least one of the operators Mt
, Nt is H.S.
then SXSY,A~B comprises all continuous linear operators from SX,A into Sy,B'
IV, SX0X,A~A comprises all continuous linear operators from SX,A into
SX,A iff for each t > 0 the operator Mt is H.S.
Proof
I. We use condition II of Theorem 4.4.
For each t > 0, a > 0 NtKMa
is a bounded operator from X into Y because
for £ sufficiently small
All operators in the last expression are bounded.
II. For each T > 0 K(T) is a H.S.-map. For £ sufficiently small, with
Properties S.l.c.
<g,Kf>y
III. Let L : SX,A ~ Sy,B be continuous. According to Theorem 4.4.11 the
operators NtLMt are bounded for each t > O. However if Nt or Mt is H.S.
for each t > 0 then NtLMt
is H.S. for each t > 0 and it defines an element in
SX®Y,A 83B' A simple verification shows that this element reproduces L by recipe (b).
IV. The if-part is a special case of III. For the only-if-part consider
the special map L =emb = r. Here 1 is the identity map. Mt lMt == M2t
can be considered as an element of SXsY,A ~A iff Mt is H.S. for all
t > O. o
CASE c: continuous linear maps: SX,A ~ Sy,B' Let P E LA . For f E SX,A we define Pf E Sy,B by
•
- 62 -
Pf E Sy, since peE) E SX0y,A~B' The definition makes sense for £
suffici iy small and does not depend on the choice of E •
~ Theorem 6.3.
I. For each P E rA the linear operator P
is continuous.
SX,A + SY,B defined by (c)
II. For each P E rA ' each f E SX,A' each G E Sy,B
<Pf,G>X = <P,f 0 G>A
III. If for each t > 0 at least one of the operators Mt , Nt is H.S. then
rA comprises all continuous linear operators from SX,A into SY,B'
IV. Consider the special case Y = X and B = A. E' A comprises all continuous linear operators from SX,A into itself iff
for each t > 0 the operator Mt 1s H.S.
Proof
I. We use condition II of Theorem 4.2.
Let f ~ 0 strongly in Sx A' For some £ > 0 M f + 0 in X. n , -E n P(£) E SX0Y,A~B' therefore there exists c > 0 such that N_cP(£) is
a bounded operator. But then N ~Pf = (N ~P(£»M f + 0 in Y, which -u n -u -€ n
shows that Pfn + 0 strongly in SY,B'
II. For Band 6 sufficiently small and positive we have
III. Let Q : SX,A + SY,B be continuous. According to Theorem 4.2.IV for each
t > 0 there exist set) > 0 such that N_S(t)QMt is a bounded map from
- 63 -
X into Y. Now because of the assumption on M , N we find that a a
WAt :: NS(t) (N_S(t)QMt) is an element of LA . It reproduces the
operator Q if the recipe (c) is applied.
IV. The if-part is a special case of III. For the only-if-part consider
the identity map 1 : SX,A ~ SX,A' In order that IMt as a function of t
is an element of LA the operator Mt has to H.S. for all t > D. 0
CASE d: Continuous linear maps: SX,A ~ Sy,S' Let ~ € LB' For F € SX,A we define ~F E Sy,B by
(d) (~F) (t) = Ili (t) M -IS: (t) F (e: (t) )
This def1ni tion makes sense for t > 0 and e: (t) > 0 sufficiently sm'lll.
(~F) (t) E: Sy,B since <P (t) E: SxeY,A lEIS' Moreover
N,(<PF) (t) = N,Ili(t)M_e:(t)F(e:(t» :: CP(t + ,)M_e:(t/(e:(t» ::: (IliF) (t + c).
Theorem 6.4.
I. For each Ili € ES the linear operator Ili
continuous.
II. For each Ili € ES' F E S~,A' g € Sy,S
<g,IliF>y
SX,A ~ Sy,B defined by (d) is
III. If for each t > 0 at least one of the operators Mt
, Nt is H.S. then
LS comprises all continuous linear operators from SX,A into Sy,Be
IV. Consider the special case Y = X and B = A. LS comprises all continuous linear operators from SX,A into itself iff
for each t > 0 the operator Mt is H.S.
Proof
1. We use Theorem 4.6.III. For each t > 0 Ntlli E SXeY,AIEB' Then according
to case a, Ntlli is a continuous linear map from SX,A into SY,B'
- 64 -
II. For a and <5 sufficiently small and positive
== <~(a) ,(I ® N ) (F e g»X Y = -a e
<9,4>F>y
III. Let ~ SX,A + Sy,B be continuous, According to Theorem 4.6.VI for each
t > 0 there exists S(t) > 0 such that Nt¥rM_S(t) is a densely defined
and bounded operator from X into Y. If one of the operators N , M is a a
H.S. for arbitrary small positive a it follows that N ¥ is H.S. fer t r
t > 0, because Nt'¥r == Nt¥r(S(t)MS(trHerelfrM_S(t) denotes the extension of Nt'¥r to the whole of X.
Since Nt'¥r == Nl.:!t(Nl.:!tljl)i-S(~t»MS(t) it follows Nt'¥r € SXeY,AIBBo Hence Nt'¥rl as a fUnction of t, belongs to ES' By recipe (d) the operator '¥ is reproduced.
IV. The if-part is a special case of III. For the only-if-part consider the
identity map 1. In order that MtI, as a function of t, can be considered
as an element in LS the operator Mt
should be H.S. for all t > O. 0
- 65 -
APPENDIX A. Functions of self-adjoint operators.
A self-adjoint operator A in a Hilbert space H admits a spectral
decomposition
00
_00
Here tA is the so-called resolution of the identity, i.e. EA is a
monotonically increasing function with values in the projection operators,
EA is strongly continuous from the right, E(_oo) = 0, E(oo) = I,
E,E a E. (' ). Let ~ be a complex valued and everywhere finite A ~ m1n A,~
Borel function on lR. On the domain
00
the operator ~(A) is defined by
00
Vx € H
where the integration is carried out with respect to the Borel measure
determined by m«A1
,A2
])
We write formally
00
It can be proved, see [YJ Ch. XI. 12, that ~(A) is a normal operator.
~(A) is self-adjoint if ~ is real valued. Further (~)X) (A) = ~(A}x(A) where the usual precautions concerning the domains must be regarded.
- 66 -
In our case A ~ 0 and the functions $ are always piece-wise continuous
and piece-wise of bounded variation. We frequently employ the notations
o < a < b ::; 00 •
a
By this we mean
-00
with
[0
1 X b(It) = a,
on (a,b]
elsewhere •
By the notation
o < b ::; 00
we mean
The elementary and elegant presentation of spectral theory in [LS] is
quite sufficient for our work. We referred to [yJ because the theory
in [LS] is not completely dressed for our purposes.
- 67 -
APPENDIX B Holomorphic semi-groups
On a Banach space X we consider a one-parameter strongly continuous
semi-group of bounded operators Pt
, t ~ 0, with the properties
P ... I, P "" P P '" P P I o t+T t T T t t ~ 0, T ~ O. Such a semi-group
describes the solutions of the convolution problem in X:
duet) = _ Au(t), dt
Here -A is the so-called infinitesimal generator of the semi-group pt
"
A is a densely defined closed linear operator which enjoys the
following property (cf. [Y] Ch. IX):
There exist constants M and a such that for every A > a the resolvent -1 n -n R).. Ie (A + AI) exists and II R).. II S M(A - a) • Then
and
Our interest is centered upon holomorphic semi-groups then Vt > a Vx E H co
Ptx E D(A ), Ptx is an infinite differentiable function of t and has
the local representation
co co ( , _ t) n
..:..:.l.1\ --:~_ An P x = n! A L Vx E X I
n=O n=O
In this case Pt has a holomorphic extension into a sector larg tl < ~
in the complex t-plane.
A necessary and sufficient condition on Pt in order to be holomorphic
is
Vx E H Vt > a Ptx E D (A)
and
3a > o Vt,O < t S 1 II tP~11 S a.
Then <p '" arctan -1
(ae) • See [yJ Ch. IX. A necessary and sufficient
- 68 -
condition on the infinitesimal generator -A to generate a holomorphic
semi-group is;
3S > a 3£ > a VA, Re A ~ 1 + £
See [y] Ch. IX.
An interesting subclass of holomorphic semi-groups on a Hilbertspace H
is generated by operators of the form A = Q + B where Q is a positive
self-adjoint operator and B is subjugated to Q in a specified sense.
See [G].
In case B = a the operators Pt
, t ~ a are all selfadjoint and Pt
has
a holomorphic extension into the whole open right half plane and a
strongly continuous extension into the closed right half plane. Whenever
t > a the operator Pt
has a self-adjoint inverse which is unbounded iff
A is unbounded. In this paper we consider only the case that A is un
bounded, self adjoint and non-negative.
- 69 -
APPENDIX C. A Banach-Steinhaus Theorem.
The following is taken from [JJ.
Theorem. Let (B,II II) be a Banach space. suppose that for every a > 0
there is given a collection A of bounded linear functionals of B a
such that 0.1
< a - A c A 2 0.
1 0.
2 statements are equivalent:
(0.1
> 0, 0.2
> 0). Then the following
Proof
It is easily seen that (ii) _ (i): if N > 0 and B > 0 are such that
I Lf I ::; N f for every fEB, LEA 6' then we can take a = 6 and M = Nil f II
in (i).
We now suppose (i), and we shall show that
yields a contradiction. To every n E ~ we can then find an fEB and an n
L E All such that I L f I > nil f II. Note that L is a bounded linear n n n n n n functional of B for every n E ~, and that (Lnf) nE~ is a bounded
sequence for every fEB. It follows therefore from the Banach-Steinhaus
theorem that there is an M > 0 such that I L f I ::; Mil f II for eve"ry n E ~ n
and every fEB. Contradiction. n
- 70 -
Acknowledgements
During the research I have profited from very stimulating discussions
with A.J.E.M. Janssen.
I wish to thank J.W. Nienhuys and J.J. Seidel for making valuable
suggestions on the presentation of the introduction.
Finally I wish to thank J. Boersma for showing me the way in Whittaker
and Watson.
- 71 -
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