ESSENTIAL SELF-ADJOINTNESS OF OPERATORS IN AN INFINITE TENSOR

PRODUCT OF HILBERT SPACES

V. A. Liskevich UDC 517.9+519.4

In this paper we study the criterion for the essential self-adjointness of series of nonnegative operators in an infinite tensor product of Hilbert spaces, expressed in terms of the weak convergence.

4 Let {J{k}~=l be a sequence of complex separable Hilbert spaces. In each ~ there is

given a densely defined closable operator A k. Let ~ = ~k be a separable subspace of k=l

the complete von Neumann product with the stabilization {Xk}k=1 ~ (for the corresponding

construction and definitions see [i]). We shall define in ~ an operator ~k:

~'--1

(loh. ~ is the linear hull of the set ~ ). Obviously, we close ~k with .~(~k). We shall

denote its closure by ~u-

In the space ~ we shall define the set

= ~,. h. {f = | f~, : f~, E E) (A'kAy,) ~ #', f~ = Xk ~k >~ p},

where p is some natural number.

It will be shown below that the weak convergence of partial sums of the series ~ A'~A~ k=1

is a sufficient and necessary condition for its essential self-adjointness on the set ~.

Definition. The vector e = ~ek, equivalent to the vector % = Xk, i.e., l<%k,e~)h-- II k = 1

<co(~.,.)~ i s t he s c a l a r p r o d u c t in / ~ ) , w i l l be c a l l e d t he v e c t o r of t he fo rm-convergence f o r the f ami ly of o p e r a t o r s {Ak}k=l ~ i f eh6~(A~) f o r ~ k = 1,2 .... and

• (A~eh, Akek)~ < oo. k = l

We shall consider in ~ the sequence of sesquilinear forms:

N N

tN [U, V] = <U, V> + ~ <~kU, ~4kV), ~9 (tN) = f-] ~(~4~). k = l k~-I

(i)

(2)

Form t N is closed, positive, symmetric, by (i) it is densely defined, and, consequently,

it has an associated self-adjoint operator B N [2]. Let us define a form t:

flu, v] = limtN [U, V], ~ (t) = {u 6 ~] ~ (tN): suptN [U] < ~}. (3) N N

N

By (i) form t is densely defined, and since t N ~ tN+ z, then by the theorem of the monotone

convergence of forms [3] it is closed, and B N + B in the sense of the strong convergence, where B is an operator associated with the form t.

XXV Congress of CPSU Automation Institute, Kiev. cheskii Zhurnal, Vol. 41, No. i, pp. 108-111, January,

April 4, 1986. ~

Translated from Ukrainskii Matemati- 1989. Original article submitted

I00 0041-5995/89/4101-0100 $12.50 �9 1989 Plenum Publishing Corporation

Let us define in the Hilbert space ~ an operator T in the following way: ~(T)=

{ / : / ~ ( T ~ ) , lim(T~f,g) e x i s t s ~g~${}, <T/,g> =lim<TJ, g>, where {Tk}k=~ ~ i s a sequence

of operators acting in S" We say that the sequence {Tk}k=~ ~ is weakly convergent in ~ to T.

THEOREM. The following conditions are equivalent:

1) ~ ~ ~ (B);

2) the sequence o f o p e r a t o r s {BN}N=I ~ i s weakly c o n v e r g e n t in ~ ;

3) B i s e s s e n t i a l l y s e l f - a d j o i n t on ~.

Remark. The i m p l i c a t i o n 3) ~ i ) i s obv ious , 1) + 3) i s a l s o s imple enough.. i n t e r e s t i n g case i s the e q u i v a l e n c e of 2) and 3) .

The most

Let us denote by PN the projection in ;

1 | 1 @... @ 1 | ~=~+~

N

~ onto the vector ; e~, PN---- ~-~NWI

LEMMA I. BN IPN~D is essentially self-adjoint ~N.

Proof. See, for example, [4, p. 473].

LEMMA 2. Let ~) c: ~) (B) and Be = (Bt~)-. Then ~ N B~I~)~'(Bo).

Proof. Let fEB~lffj. Then by Lemma 1 there exists a sequence {/m}~=1 ~P~D such that

fm + f and BNfm + BNf. Since[m~9, then it is sufficient to show that the limit limB/m exists. Since ~P~, then, without loss of generality it is enough to consider ~ectors

N , of the form ]m=~| ~ e~, f~m~a.s. | D. Then

k = N + l k~l " )~

= (}2 r2'| | |

The f i r s t component i s c o n v e r g e n t , t he l a t t e r does no t depend on m, hence ~i~m Bf m e x i s t s .

Remark. The p r o o f o f Lemma 2 i s a n a l o g i c a l t o t he r e a s o n i n g in [1, p. 323] .

LEMMA 3. The following conditions are equivalent:

1) ~ ( B ) ;

2) the sequence of operators {BN} is weakly convergent in the space ~ with the norm

I1" tl' = II B-"% I1~- N

Proof. It is clear that eE~)(BI/2), eE,.q)(Bu). Let ~pE~(Bi"~). Then <B2ve,~)=E(~e, k=I

~h~p)-+(BZ/2e, BW~cp), N-..~oo. Suppose t h a t c o n d i t i o n 1) i s s a t i s f i e d , i . e . , eEff)(B). Then ~tpE$3(BU2)(Btve,. cp).-~(Be, cp), N.-.~.oo. Let now 2) be s a t i s f i e d . Then (B2ve, cp>--*(e',~L i . e . , <e',~>=(Bl/~e, BI/2q~). By the representation theorem [2] we have eE~)(B).

Proof of the Theorem. At first we shall show that in Lemma 3 condition 2) can be re- placed by the weak convergence of the sequence {BN} in ~.

Let C k be an operator associated with the form (~,u,~k~) (C k is self-adjoint and non-

negative). Since C~----!)~/(l~l~... ~fk(k)|174 then the operators C k have commuting spec-

tral solutions, so that there exists a decomposition of unity EX and functions Vk(%), such

that ~k(k) e 0 and C~=~wk(k)dE z [5]. We have

N N

k ~ l ~ 1

By the Levy theorem we can pass to the limit in (4) for N ~

I01

'E cj,:>= S E Now it is obvious that for any f, satisfying the condition

(5)

the following equality holds

k = l k = l

Notice now that B 0 c B, hence B 0 com~nutes with B N. Then, obviously, we have

II Bo (B;' - - B~ ~): II = II ( B ~ ~ - - B ~ ~ ) B J [I --~ 0, M , N ~ ~ , "V'f E ~ .

By the closedness of the operator Bo B-I~(Bo) and BoB~I~-*-BoB-~: ~E~. To conclude

the proof it remains to notice that (B~B-I~) - =(BIB-I~) - =B (cf. [6]).

LITERATURE CITED

.

2. 3.

4.

5. 6.

Yu. M. Berezanskii, Self-Adjoint Operators in Spaces of Functions of Infinite Number of Variables [in Russian], Naukova Dumka, Kiev (1978). T. Kato, Perturbation Theory of Operators [Russian translation], Mir, Moscow (1972). B. Simon, "A canonical decomposition for quadratic forms with applications to monotone theorems," J. Funct. Anal., 28, No. 3, 377-385 (1978). Yu. M. Berezanskii, Eigenfunction Expansion of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965). A. I. Plesner, Spectral Theory of Linear Operators [in Russian], Nauka, Moscow (1965). I. M. Burban, "On the weak convergence in infinite tensor products of Hilbert spaces," Teor. Mat. Fiz., 9, No. 3, 318-322 (1971).

HYPERBOLIC EQUATION METHOD IN LP-SPACES

M. A. Perel'muter UDC 517.98

Introduction. The hyperbolic equation method arose in Povzner's work [i] and was fur- ther developed in the work of Levitan, Berezanskii, Orochko, Chernov, Kato and others (see, for example, [2-4]. This method turned out to be an effective means of studying the essen-

l

0 0 L~ tial self-adjointness of elliptic operators of the form H =m 0--~-h ahj ~j in (~l,dLx)

k. l= l

on a set consisting of functions with compact support. The main advantage of the given method is that it allows us to obtain results without any assumptions on the smoothness of the coefficients (see, for example, [5]). This method is based on the fact that under cer- tain bounds on the growth of the coefficients akj(x) as ]xl-~oo cos(fl/-H) L~(~l,d~x)~--L~(~ ;, fx), where L~(N:,d~x) is the subspace of L~(NL,d~), consisting of functions with compact

support.

In the Banach space LP(R I, fx) the natural analog of the essential self-adjointness of a nonnegative operator is the fact that the closure of the operator is a generator of a strongly continuous semigroup. Applying the hyperbolic equation method in this situation

Ukrainskii Nil of Design of Steel Construction. Zhurnal, Vol. 41, No. i, pp. 111-116, January 1989. 1986; revision submitted August 6, 1986.

Translated from Ukrainskii Matematicheskii

Original article submitted January 27,

102 0061-5995/89/4101-0102 $12.50 �9 1989 Plenum Publishing Corporation