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Summability, Matrix Transformations and TheirApplications

Doctoral School, Tartu University, Tartu, Estonia

13 October, 2011

Eberhard Malkowsky

Department of MathematicsFaculty of Arts and Sciences

Fatih University34500 Buyukcekmece, Istanbul

Turkey

email: ema@Bankerinter.net

Eberhard.Malkowsky@math.uni-giessen.de

1Research supported by the research project #174025 of the Serbian Ministry of Science, Technology and Environment

Mathematics Subject Classification: 46A45, 40C05

Key words and phrases: Sequence spaces, dual spaces, matrix transformations, Hausdorff measure of noncompactness, compact operators, Fredholm operators

Contents

1 Introduction

2 Matrix Transformations

2.1 Notations

2.2 FK, BK and AK spaces

2.3 The dual spaces

2.4 General results

3 Compact Operators

3.1 The Hausdorff measure of noncompactness

4 Fredholm Operators on c0

5 The Spaces of Strongly Summable and Bounded Sequences

6 Graphical Representations of Neighbourhoods

7 Wulff’s Crystals and Potential Surfaces

References

1 Introduction

The theory of matrix transformations is a wide field in summability; it dealswith the characterisations of classes of matrix mappings between sequencespaces by giving necessary and sufficient conditions on the entries of theinfinite matrices.

The most important applications are

• inclusion, Mercerian and Tauberian theorems

• characterisation of compact linear operators

• study of Fredholm operators given by matrices

• determination of Banach algebras of matrix transformations

• spectral theory, invertibility of operators, solvability of infinite systemsof linear equations

Summability

The classical summability theory deals with a generalization of convergenceof sequences and series. One original idea was to assign a limit to divergentsequences or series.

Classical methods of summability were applied to problems in analysis suchas the analytic continuation of power series and improvement of the rateof convergence of numerical series.

This was achieved by considering a transform, given by an infinite matrix,rather than the original sequence or series; most popular are

• Hausdorff matrices, and their special cases

– Cesaro matrices Cα

– Euler matrices Eq

– Holder matrices Hα

• Norlund matrices (N, p)

Concepts of summability

Let A = (ank)∞n,k=1 be an infinite matrix, x = (xk)∞k=1 be a sequence, e =

(1, 1, 1 · · · ), Anx =∑∞

k=1ankxk and Ax = (Anx)∞n=1 be the sequence ofthe A transforms of x. There are three concepts of summability.

• Ordinary summability : x is summable A if

limn→∞

Anx = ℓ for some ℓ ∈ |C

• strong summability : x is strongly summable A with index p > 0 if

limn→∞

An(|x − ℓ · e|p) = limn→∞

∞∑

ank|xk − ℓ|p = 0 for some ℓ ∈ |C

• absolute summability : x is absolutely summable A with index p > 0 if∞∑

|Anx − An−1x|p < ∞.

An example

Example 1.1 Let the matrix A be given by ank = 1/n for 1 ≤ k ≤ n andank = 0 for k > n (n = 1, 2, . . . ). Then the A transforms of the sequencex are the arithmetic means of the terms of x, that is,

σn =1

and A defines the Cesaro method C1 of order 1.

• Every convergent sequence is summable C1 and the limit is preserved

• the divergent sequence ((−1)k)∞k=1 is summable C1 to 0

• strong summability of index 1 implies ordinary summability to the samelimit; the converse is not true, in general

• absolute summability with index 1 implies ordinary summability.

2 Matrix Transformations

Let X and Y be subsets of the set ω of all complex sequences. The theoryof matrix transformations deals with the characterisation of the class (X, Y )of all infinite matrices that map X into Y . So A ∈ (X, Y ) if and only if

(2.1) Anx converges for all n and all x ∈ X

(2.2) Ax ∈ Y for all x ∈ X .

We are going to study

• FK, BK and AK spaces

• dual spaces

• special classes of matrix transformations

2.1 Notations

ℓ∞

the sets of all

bounded

convergent

finite

sequences x = (xk)∞k=1

e(n) (n ∈ IN)

the sequences with

ek = 1, (k = 1, 2, . . . )

e(n)k =

1 (k = n)

0 (k 6= n)

BX(r, x0), SX(r, x0) and X ′

Let (X, d) be a metric space. Then

• BX(r, x0) = {x ∈ X : d(x, x0) < r} is the open ball andSX(r, x0) = {x ∈ X : d(x, x0) = r} is the sphere in X of radius r > 0and centre in x0 ∈ X ; we write BX = BX(1, 0) and SX = SX(1, 0)

• if X is a linear metric space then X ′ is the continous dual of X , thatis, the set of all continuous linear functionals on X .

B(X, Y ) and X∗

Let X and Y be normed spaces. Then

• B(X, Y ) is the space of all bounded linear operators L : X → Y with

‖L‖ = supx∈SX‖L(x)‖

• X∗ = B(X, |C) is the continuous dual of X with

‖f‖ = supx∈SX|f (x)| for all f ∈ X∗

2.2 FK, BK and AK spaces

The theory of FK, BK and AK spaces from modern functional analysisis the most powerful tool in the characterisation of matrix transformations.The reason is that matrix transformations between FK spaces are contin-uous.

The theory, however, fails in some cases, for instance, when the initialsequence space has no Schauder basis, as in the determination of the class(ℓ∞, c) of coercive matrices.

In such cases, the method of the gliding hump from classical analysis isapplied.

FK and BK spaces ([27, p. 55]

A subspace X of ω is said to be an FK space if it is a Frechet space,that is, a complete, locally convex, linear metric space, with continuouscoordinates

Pn : X → |C (n = 1, 2, . . . ) where Pn(x) = xn;

a BK space is an FK space with its metric given by a norm.

AK property [27, Definition 4.2.13]

An FK space X ⊃ φ is said to have AK if every sequence x = (xk)∞k=1 ∈X has a unique representation

∞∑

xke(k), that is, x = lim

n→∞

xke(k).

(ω, dω)

Example 2.2 The set ω is a Frechet space with ([26, Example 11.3.1])

(2.3) dω(x, y) =

∞∑

|xk − yk|

1 + |xk − yk|for all x, y ∈ ω

(i) convergence in (ω, dω) is equivalent to coordinatewise convergence

([26, Theorem 4.1.1 and Example 4.1.5]).So (ω, dω) is an FK space, which obviously has AK.

Remark 2.3 In view of (i), an FK space X is a

(ii) Frechet sequence space with its metric stronger than the metric dω ofω on X ,

or equivalently,

(iii) a Frechet sequence space which is continuously embedded in ω.

The FK spaces ℓ(p) and c0(p)

Example 2.4 Let p = (pk)∞k=1 be a bounded sequence of positive realswith M = max{1, supk pk}. Then

ℓ(p) = {x ∈ ω :∑∞

k=1|xk|pk < ∞}

andc0(p) = {x ∈ ω : lim

k→∞|xk|

pk = 0}

are FK spaces with AK with

d(p)(x, y) =

∞∑

|xk − yk|pk

([15, Theorem 1])

and d0,(p)(x, y) = supk

|xk − yk|pk/M ([18, Theorem 2]).

The classical BK spaces

Example 2.5 (a) The sets ℓp (1 ≤ p < ∞), c0, c and ℓ∞ are BK spaceswith

‖x‖p =

∞∑

and ‖x‖∞ = supk

|xk| ([26, Example 11.3.1]);

c0 is a closed subspace of c and c is a closed subspace of ℓ∞ ([27, Corollary4.2.4]); ℓp and c0 have AK ([27, Example 7.3.7]);

every sequence x = (xk)∞k=1 ∈ c has a unique representation

x = ξ · e +

∞∑

(xk − ξ)e(k) where ξ = limk→∞

xk ([17, Example 3.4.11]);

ℓ∞ has no Schauder basis, since it is not separable ([17, Theorem 3.4.7,Problem 3.4.4]).

2.3 The dual spaces

If X, Y ⊂ ω, then

M(X, Y ) ={

a ∈ ω : ax = (akxk)∞k=1 ∈ Y for all x ∈ X}

is called multiplier space of X and Y;

the sets

Xα = M(X, ℓ1), Xβ = M(X, cs) and Xγ = M(X, cs)

are called the α–, β– and γ–duals of X .

Properties of duals of BK spaces

Theorem 2.6 ([27, Theorem 4.3.15]) Let X and Y be BK spaces. ThenM(X, Y ) is a BK space with

‖z‖ = sup{‖xz‖ : ‖x‖ = 1} (z ∈ M(X, Y ));

in particular, the Xα, Xβ, and Xγ are BK spaces.

Theorem 2.6 does not extend to FK spaces, in general

Example 2.7 The space (ω, dω) is an FK space (Example 2.2) and ωβ =φ, as we will see in Example 2.10 (i), but φ has no Frechet topology ([27,4.0.5]).

Relation between X′and Xβ

Theorem 2.8 (a) If X ⊃ φ is an FK space with AK then Xβ = Xγ

([27, Theorem 7.2.7 (iii)]).

(b) ([27, Theorem 7.2.9]) Let X ⊃ φ be an FK space.

Then Xβ ⊂ X′in the sense that each sequence a ∈ Xβ can be used to

represent a function fa ∈ X ′ with fa(x) =∑∞

k=1akxk for all x ∈ X , and

the map T : Xβ → X ′ with T (a) = fa is linear and one to one.If X has AK then T is an isomorphism.

The norm ‖ · ‖∗X

Let X ⊂ ω be a normed sequence space, and a ∈ ω. Then we write

‖a‖∗X = supx∈SX

∞∑

provided the expression on the righthand side exists and is finite which isthe case whenever X is a BK space and a ∈ Xβ by Theorem 2.8 (b).

Norm isomorphic, ≡

If X and Y are norm isomorphic then we write

X ≡ Y .

The β–duals of ℓ(p) and c0(p)

Example 2.9 (a) If pk > 1 and qk = pk/(pk − 1) for all k, then

(ℓ(p))β =⋃

a ∈ ω :

∞∑

qk< ∞

([16, Theorem 11]).

(b) If p is any sequence of positive reals then,

(c0(p))β =⋃

a ∈ ω :

∞∑

|ak| · N−1/pk < ∞

([16, Theorem 6]).

The duals of the classical BK spaces

Example 2.10 ([26, Examples 6.4.2, 6.4.3 and 6.4.4]) We have

(i) ωβ = φ and φβ = ω

(ii) ℓ∗1 ≡ ℓβ1 = ℓ∞; ℓ∗p ≡ ℓ

βp = ℓq for 1 < p < ∞ and q = p/(p − 1)

(iii) c∗0 ≡ cβ0 = cβ = ℓ

β∞ = ℓ1

(iv) f ∈ c∗ if and only if

f (x) = χf · limk→∞

xk +∞∑

k=1akxk where a = (f (e(k)))∞k=1 ∈ ℓ1 and

χf = f (e) −∞∑

k=1f (e(k)), and ‖f‖ = |χf | + ‖a‖1

(v) ℓ∗∞ is not given by any sequence space ([26, Example 6.4.8]), but

(2.4) ‖ · ‖∗ℓ∞ = ‖ · ‖1 on ℓβ∞.

2.4 General results

We list a few useful known results.

The relation between B(X, Y ) and (X, Y )

Theorem 2.11 Let X and Y be BK spaces.

(a) Then we have (X, Y ) ⊂ B(X, Y ); this means that if A ∈ (X, Y ), thenLA ∈ B(X, Y ) where

LA(x) = Ax (x ∈ X) ([27, Theorem 4.2.8]).

(b) If X has AK then we have B(X, Y ) ⊂ (X, Y ); this means everyL ∈ B(X, Y ) is given by a matrix A ∈ (X, Y ) such that

L(x) = Ax (x ∈ X) ([12, Theorem 1.9]).

The class (X, ℓ∞) ([20, Theorem 1.23 (b)])

Theorem 2.12 Let X be a BK space. Then A ∈ (X, ℓ∞) if and only if

‖A‖(X,ℓ∞) = supn

‖An‖∗X < ∞;

moreover, if A ∈ (X, Y ) then ‖LA‖ = ‖A‖(X,ℓ∞).

Improvement of mapping

Theorem 2.13 (a) ([27, 8.3.6]) Let X be an FK space with φ = X , Yand Y1 be FK spaces with Y1 a closed subspace of Y . Then we have

A ∈ (X, Y1) if and only if A ∈ (X, Y ) and Ae(k) ∈ Y1 for all k.

(b) ([27, 8.3.7]) Let X be an FK space and X1 = X ⊕ e. Then

A ∈ (X1, Y ) if and only if A ∈ (X, Y ) and Ae ∈ Y .

Mapping properties of the transpose

Let At denote the transpose of the matrix A.

Theorem 2.14 ([27, Theorem 8.3.9]) Let X and Z be BK spaces withAK and Y = Zβ. Then we have

(a) (Xββ, Y ) = (X, Y );

(b) A ∈ (X, Y ) if and only if At ∈ (Z, Xβ).

The classes (X, Y ) for X = ℓ∞, c, c0

Example 2.15 We have(a) A ∈ (ℓ∞, ℓ∞) if and only if

(2.5) ‖A‖(∞,∞) = supn

∞∑

|ank| < ∞ ([27, 8.4.5A]);

if A ∈ (ℓ∞, ℓ∞) then

(2.6) ‖LA‖ = ‖A‖(∞,∞)

(b) (c0, ℓ∞) = (c, ℓ∞) = (ℓ∞, ℓ∞) ([27, 8.4.5A]).

Proof. (a) Since ℓ∞ is a BK space by Example 2.5 (a), the statementsfollow from Theorem 2.12 and (2.4).

(b) Since c0 is a BK space with AK by Example 2.5 (a), and cββ0 = ℓ

ℓ∞ by Example 2.10 (iii) and (ii), Theorem 2.14 (a) yields (c0, ℓ∞) =(ℓ∞, ℓ∞); this and c0 ⊂ c ⊂ ℓ∞ yield (c, ℓ∞) = (ℓ∞, ℓ∞). �

The class B(ℓ1, ℓ1)

Example 2.16 ([20, Theorem 2.27]) We have B(ℓ1, ℓ1) = (ℓ1, ℓ1) by The-orem 2.11 and Example 2.5 (b). Also A ∈ (ℓ1, ℓ1) if and only if

(2.7) ‖A‖(1,1) = supk

∞∑

|ank| < ∞

moreover, if L ∈ B(ℓ1, ℓ1) then

(2.8) ‖L‖ = ‖A‖(1,1).

Proof. We apply Theorem 2.14 (b) with X = ℓ1 and Z = c0, BK spaces

with AK by Example 2.5, and Y = cβ0 = ℓ1 by Example 2.10 (iii) to obtain

A ∈ (ℓ1, ℓ1) if and only if At ∈ (ℓ∞, ℓ∞), and this is the case by (2.5) ifand only if

‖At‖(∞,∞) = supn

∞∑

|akn| = ‖A‖(1,1) < ∞.

Furthermore, if L ∈ (ℓ1, ℓ1), then

‖L(x)‖1 = ‖Ax‖1 =

∞∑

|Anx| =

∞∑

≤∞∑

∞∑

≤ ‖A‖(1,1) · ‖x‖1

implies

(2.9) ‖L‖ ≤ ‖A‖(1,1).

We also have for e(k) ∈ Sℓ1(k ∈ IN)

‖L(e(k))‖1 = ‖Ae(k)‖1 =

∞∑

∣Ane(k)

∞∑

|ank| ≤ ‖L‖,

hence ‖A‖(1,1) ≤ ‖L‖. This and (2.9) imply (2.8). �

Regular matrices, the Silverman–Toeplitz theorem

A matrix A ∈ (c, c) is said to be regular if limn→∞Anx = limk→∞ xkfor all x ∈ c.

Theorem 2.17 (Silverman–Toeplitz, 1911)(a) ([27, Theorem 1.3.6]) We have A ∈ (c, c) if and only if

(i) ‖A‖(∞,∞) = supn

∞∑

|ank| < ∞ ((2.5)),

(ii) αk = limn→∞

ank exists for each k ∈ IN,

(iii) α = limn→∞

∞∑

ank exists.

(b) Let A ∈ (c, c) and x ∈ c. Then we have

(2.10) limn→∞

α −

∞∑

· limk→∞

∞∑

αkxk.

(c) A matrix A is regular if and only if (i) holds and

(ii’) limn→∞

ank = 0 for each k ∈ IN,

(iii’) limn→∞

∞∑

ank = 1.

Proof. Part (a) follows from Theorem 2.13 (a) and (b) and Example 2.15.(b) This is elementary.Part (c) follows from Parts (a) and (b). �

Schur’s theorem; proof by the method of the gliding hump

Theorem 2.18 (Schur) We have(a) ([27, Theorem 1.7.18]) A ∈ (ℓ∞, c) if and only if

(2.11)

∞∑

|ank| converges uniformly in n

(2.12) limn→∞

ank = αk exists for each k ∈ IN;

(b) ([27, Theorem 1.7.19]) A ∈ (ℓ∞, c0) if and only if condition (2.11)holds and

(2.13) limn→∞

ank = 0 for each k ∈ IN.

Remark 2.19 The conditions in (2.11) and (2.13) are equivalent to

limn→∞

∞∑

|ank| = 0 ([25, 21. (21.1)]).

Applications

Example 2.20 (Steinhaus) A regular matrix cannot sum all bounded se-quences.

Proof. If there were a matrix regular A ∈ (ℓ∞, c), then it would followfrom the conditions in (2.11) of Theorem 2.18 and in (ii’) and (iii’) in Part(c) of Theorem 2.17 that

1 = limn→∞

∞∑

limn→∞

ank = 0, a contradiction. �

Example 2.21 Schur’s theorem can be applied to show that weak andstrong convergence coincide in ℓ1.

Proof of Example 2.21

We assume that the sequence (x(n))∞n=1 is weakly convergent to x in ℓ1,that is,

f (x(n)) − f (x) → 0 (n → ∞) for each f ∈ ℓ∗1.

To every f ∈ ℓ∗1 there corresponds a sequence a ∈ ℓ∞ (Example 2.10 (ii))such that

f (y) =

∞∑

akyk for all y ∈ ℓ1.

We define the matrix B = (bnk)∞n,k=1 by bnk = x(n)k −xk (n, k = 1, 2, . . . ).

Then we have for all a ∈ ℓ∞

f (x(n)) − f (x) = f (x(n) − x) =

∞∑

ak(x(n)k − xk)

∞∑

bnkak → 0 (n → ∞),

that is, B ∈ (ℓ∞, c0). It follows from Theorem 2.18 (b), that∑∞

k=1|bnk|converges uniformly in n and limn→∞ bnk = 0 for each k. Thus we have

‖x(n) − x‖1 =

∞∑

|x(n)k − xk| =

∞∑

|bnk| → 0 (n → ∞). �

3 Compact Operators

The Hausdorff measure of noncompactness is most effective in determiningconditions for a linear operator to be compact.

Measures of noncompactness are also very useful tools in the theory ofoperator equations in Banach spaces, in particular in fixed point theory.

The first measure of noncompactness, denoted by α, was defined and stud-ied by Kuratowski [13] in 1930. Later, in 1955, Darbo [6] used the functionα to prove a generalisation of Schauder’s fixed point theorem to noncom-pact operators.

Other measures of noncompactness, the Hausdorff measure of noncompact-ness, were introduced by Goldenstein, Gohberg and Markus [9] in 1957,and later studied by Goldenstein and Markus [10] in 1968, and by Istratesku[11] in 1972.

The general theory of measures of noncompactness can be found in [1, 2].

Notations

We write MX for the class of all bounded subsets of a metric space (X, d).

Let X and Y be infinite–dimensional complex Banach spaces.A linear operator L : X → Y is said to be compact if

• DL = X for the domain DL of L

• for every bounded sequence (xn) in X , the sequence (L(xn)) has aconvergent subsequence.

We write C(X, Y ) for the class of all compact operators from X into Y .

A norm ‖ · ‖ on a sequence space is said to be monotone, if

x, x′ ∈ X with |xk| ≤ |x′k| for all k implies ‖x‖ ≤ ‖x′‖.

3.1 The Hausdorff measure of noncompactness

We introduce the notion of a measure of noncompactness.

Definition 3.1 ([1, Definition II.1.1]) Let X be a metric space. A map

µ : MX → [0,∞)

is said to be a measure of noncompactness on X (MNC) if it satisfiesthe following properties

(MNC.1) µ(Q) = 0 if and only if Q is relatively compact (Regularity)

(MNC.2) µ(Q) = µ(Q) for all Q ∈ MX (Invariance under closure)

(MNC.3) µ(Q1 ∪ Q2) = max{µ(Q1), µ(Q2)} for all Q1, Q2 ∈ MX

(Semi–additivity)

Properties of MNC’s in metric spaces

Theorem 3.2 ([20, Lemma 2.11] or [1, (1),(2),(3), p. 19])Let µ be a MNC in a metric space X .

Then we have for all Q, Q1, Q2 ∈ MX

(MNC.1’) Q1 ⊂ Q2 implies µ(Q1) ≤ µ(Q2) (Monotonicity)

(MNC.2’) µ(Q1 ∩ Q2) ≤ min{µ(Q1), µ(Q2)}

(MNC.3’) µ(Q) = 0 for every finite set Q (Non–singularity)

Properties of MNC’s in normed spaces

Theorem 3.3 ([20, Theorem 2.12] or [1, (6),(5),(7), p. 19])Let µ be a MNC in a normed space X .

Then we have for all Q, Q1, Q2 ∈ MX and all scalars λ

(MNC.4’) µ(Q1 + Q2) ≤ µ(Q1) + µ(Q2)

(Algebraic semi–additivity)

(MNC.5’) µ(λQ1) = |λ|µ(Q1) (Homogenity)

(MNC.6’) µ(x + Q) = µ(Q) (Translation invariance)

Hausdorff MNC of bounded sets

Definition 3.4 ([20, Definition 2.10] or[1, Definition II.2.1])Let (X, d) be a metric space and Q ∈ MX .

The Hausdorff measure of noncompactness (HMNC) of Q isdefined by

χ(Q) = inf

ǫ > 0 : Q ⊂

B(ri, xi); ri < ǫ, xi ∈ X

Invariance of the HMNC under the passage to the convexhull

Proposition 3.5 ([20, Theorem 2.12 (2.31)] or [1, Theorem II.2.4])Let X be a normed space. Then we have for all Q ∈ MX

χ(co(Q)) = χ(Q) where co(Q) is the convex hull of Q.

HMNC of the unit ball in a normed space

Theorem 3.6 ([20, Theorem 2.14]) or [1, Theorem II.2.5])Let X be an infinite dimensional normed space. Then we have

(3.1) χ(BX) = 1 .

HMNC of bounded sets in BK spaces

Theorem 3.7 ([3, Theorem 3.4]) (a) Let X be a monotone BK spacewith AK and Pn : X → X be the projectors onto the linear span of{e(1), e(2), . . . , e(n)} for n = 1, 2, . . . . Then

(3.2) χ(Q) = limn→∞

supx∈Q

‖(I − Pn)(x)‖

for all Q ∈ MX .

(b) Let Pn : c → c be the projectors onto the linear span of {e, e(1),

. . . , e(n)} for n = 1, 2, . . . .Then

(3.3)1

2· limn→∞

supx∈Q

‖(I − Pn)(x)‖

≤ χ(Q) ≤

limn→∞

supx∈Q

‖(I − Pn)(x)‖

for all Q ∈ MX .

HMNC of operators between Banach spaces

Definition 3.8 ([20, Definition 2.24])Let X and Y be Banach spaces and χ1 and χ2 be HMNC’s on X and Y .Then the operator L : X → Y is called (χ1, χ2)-bounded if

• L(Q) ∈ MY for every Q ∈ MX

• there exists a constant C > 0 such that

(3.4) χ2(L(Q)) ≤ C · χ1(Q) for every Q ∈ MX .

If an operator L is (χ1, χ2)- bounded then the number

‖L‖(χ1,χ2)= inf{C > 0 : (3.4) holds}

is called the (χ1, χ2)- measure of noncompactness of L.

If χ1 = χ2 = χ, then we write ‖L‖χ = ‖L‖(χ,χ).

Properties of the HMNC of operators

Theorem 3.9 ([20, Theorem 2.25]) Let X and Y be Banach spaces andL ∈ B(X, Y ). Then

(3.5) ‖L‖χ = χ(L(SX)) = χ(L(BX)).

Theorem 3.10 ([20, Corollary 2.26]) Let X , Y and Z be Banach spaces,L ∈ B(X, Y ) and L ∈ B(Y, Z). Then ‖ · ‖χ is a seminorm on B(X, Y )and

‖L‖χ = 0 if and only if L ∈ C(X, Y ),(3.6)

‖L‖χ≤ ‖L‖,

‖L + K‖χ= ‖L‖χ for each K ∈ C(X, Y ),

‖L ◦ L‖χ≤ ‖L‖χ · ‖L‖χ.

‖L‖χ for L ∈ B(ℓ1, ℓ1)

Example 3.11 (Goldenstein, Gohberg and Markus)([20, Theorem 2.28]) Let L ∈ B(ℓ1, ℓ1). Then we have by Example 2.16,(2.7), (2.8), (3.2) and (3.5)

(3.7) ‖L‖χ = ‖LA‖χ = limm→∞

∞∑

Example 3.12 (Goldenstein, Gohberg and Markus)([20, Corollary 2.29]) Let L ∈ B(ℓ1, ℓ1). Then we have by (3.6) and (3.7),L ∈ C(ℓ1, ℓ1) if and only if

(3.8) limm→∞

∞∑

Compact matrix operators between the classical sequencespaces

Remark 3.13 (a) It is clear from (3.8) that I ∈ B(ℓ1, ℓ1) \ C(ℓ1, ℓ1).

(b) The necessary and sufficient conditions for an operator given by a matrixA ∈ (X, Y ) to be compact can be found in [23, (a)–(b), p. 85] when

X = ℓ1, ℓ∞, c0, c, Y = ℓr for 1 ≤ r < ∞;

X = ℓp for 1 < p ≤ ∞, Y = ℓ1, ℓ∞;

X = c0, Y = ℓ∞.

(c) The class C(ℓ1, ℓ∞) cannot be determined by the use of the Hausdorffmeasure of noncompactness. The characterisation of the class C(ℓ1, ℓ∞) isestablished in [23, Theorem 5].

An estimate for ‖L‖χ when L ∈ B(X, c)

Theorem 3.14 ([3, Theorem 3.5]) Let X be a BK space with AK.

Then every L ∈ B(X, c) can be represented by a matrix A = (ank)∞n,k=1 ∈

(X, c) such that L(x) = Ax for all x ∈ X (Theorem 2.11 (b)).

The Hausdorff measure of noncompactness of L satisfies

(3.9)1

2· limr→∞

supn≥r

‖An − α‖∗X

≤ ‖L‖χ ≤

limr→∞

supn≥r

‖An − α‖∗X

(3.10) αk = limk→∞

ank for every k ∈ IN and (αk)∞k=1 ∈ Xβ.

Representation of L ∈ B(c, c)

Lemma 3.15 Every L ∈ B(c, c) is given by a matrix B = (bnk)∞n=1,k=0such that

L(x) = (bn0 · ξ +∑∞

k=1bnkxk)∞n=1 for all x ∈ c, where ξ = limk→∞

(3.11) βk = limn→∞

bnk exists for all k ≥ 1, β = limn→∞

∞∑

bnk exists,

‖L‖ = supn∈N

∞∑

Furthermore, we have

η = limn→∞

(L(x))n =(

β −∑∞

k=1βk

ξ +∑∞

k=1βkxk for all x ∈ c.

A formula for ‖L‖χ when L ∈ B(c, c)

Theorem 3.16 ([19, Theorem 1]) If L ∈ B(c, c), then we have

2· limn→∞

bn0 − β +

∞∑

|bnk − βk|

≤ ‖L‖χ ≤ limn→∞

bn0 − β +

∞∑

|bnk − βk|

where bnk (n ∈ IN; k ∈ IN0) are the entries of the matrix B that representsL by Lemma 3.15,and β and βk (k ∈ IN) are given by (3.11).

Characterisation of C(X, Y ) when X, Y = c0, c

Corollary 3.17 Let L ∈ B(X, Y ). Then the necessary and sufficient con-ditions for L ∈ C(X, Y ) can be read from the table

FromTo

c0 1. 2.c 3. 4.

1. limr→∞

(supn≥r∑∞

k=1 |ank|) = 0;

2. limn→∞

(∑∞

k=0 |bnk|) = 0;

3. limr→∞

(supn≥r∑∞

k=1 |ank − αk|) = 0;

4. limn→∞

∣bn0 − β +∑∞

k=1 βk

∣ +∑∞

k=1 |bnk − βk|)

Compact matrix operators in (c, c0) and (c, c)

Corollary 3.18 (a) Let A ∈ (c, c0). Then we have LA ∈ C(c, c0) if andonly if

limn→∞

∞∑

|ank| = 0.

(b) Let A ∈ (c, c). Then we have LA ∈ C(c, c) if and only if

limn→∞

∞∑

αk − α

∞∑

|ank − αk|

with αk (k ∈ IN) from (3.10) or (ii) in Theorem 2.17, and

α = limn→∞

∞∑

ank ((iii) in Theorem 2.17).

Regular operators

An operator L ∈ B(c, c) is said to be regular, if

limn→∞Ln(x) = ξ for all x ∈ c, where ξ = limk→∞

Corollary 3.19 ([5, Corollary 3]) Let L ∈ B(c, c) be regular. Then wehave L ∈ C(c, c) if and only if

(3.12) limn→∞

(|bn0 − 1| +∑∞

k=1|bnk|) = 0

with bnk (n ∈ IN; k ∈ IN0) from Theorem 3.16.

Remark 3.20 If A is a regular matrix then LA cannot be compact, sincewe have bn0 = 0 for all n ∈ IN and

1 +∑∞

k=1|ank| ≥ 1 6= 0 for all n,

and so (3.12) in Corollary 3.19 cannot hold.

4 Fredholm Operators on c0

Here we apply our previous results to establish sufficient conditions for amatrix operator in c0 to be a Fredholm operator. The presented results arethe special cases T = I of those in [7]

If we denote the set of finite rank operators by F(X, Y ), and suppose thatX is a normed space and Y is a Banach space, then it is well known thatF(X, Y ) ⊂ C(X, Y ) ([22], p.111). In particular, if X is a Banach spaceand Y is a Hilbert space, then the set of compact operators is the closure ofthe set of finite rank operators, that is, F(X, Y ) = C(X, Y ) ([22], p.111).But, here, we deal with Banach spaces.

The concept of the finite dimensional case connects compact and Fredholmoperators. Here, we will use a result based on the relation between themand compact operators.

Fredholm operators

Definition 4.1 Let X and Y be Banach spaces and L ∈ B(X, Y ). Wedenote the null and the range spaces of L by N(L) and R(L). Then L issaid to be a Fredholm operator if the following conditions hold:

(1) N(L) is finite dimensional;

(2) Y/R(L) is finite dimensional.

The set of Fredholm operators from X to Y is denoted by Φ(X, Y ); wewrite Φ(X) = Φ(X, X).

Remark 4.2 The range of a Fredholm operator is closed.

Known result

Theorem 4.3 ([24, p.106]) Let X be a Banach space and L ∈ B(X) =B(X, X), then I − L ∈ Φ(X) where I is the identity operator on X .

A sufficient condition for B(X) ⊂ Φ(X)

Theorem 4.4 Let L ∈ B(c0). If

(4.1) limr→∞

supn≥r

|1 − ann| +

∞∑

k=1,k 6=n

then we have L ∈ Φ(c0).

Proof. We put C = I − A where A is the matrix with Ax = L(x) for allx ∈ c0 (Theorem 2.11 (b)). Then LC ∈ C(c0, c0) by 1. in Corollary 3.17if and only if

limr→∞

supn≥r

∞∑

which is (4.1). Now the statement follows with Theorem 4.3. �

5 The Spaces of Strongly Summable andBounded Sequences

Let 1 ≤ p < ∞ throughout

x ∈ ω : limn→∞

|xk − ξ|p

= 0 for some ξ ∈ |C

wp∞ =

x ∈ ω : supn

Strong limit

If x ∈ wp then the strong C1–limit of x is the number ξ ∈ |C with

(5.1) limn→∞

|xk − ξ|p

Sectional and block norms

We define the sectional ‖ · ‖s and block norms ‖ · ‖b by

(5.2) ‖x‖s = supn

and ‖x‖b = supν

2ν+1−1∑

Topological properties of wp0, wp and w

Theorem 5.1 (a) The strong limit is unique for each x ∈ wp ([14]).

(b) We have wp0 ⊂ wp ⊂ w

p∞; ‖ · ‖s and ‖ · ‖b are equivalent norms on

wp0, wp and w

p∞ ([21, Proposition 5.3]);

(c) wp0, wp and w

p∞ are BK spaces with ‖ · ‖b ([21, Proposition 5.3]);

wp0 is a closed subspace of wp, and wp is a closed subspace of w

p∞ (Part

(b) and [27, Corollary 4.2.4]);

wp0 has AK ([21, Remark 5.4 (b)]); every sequence x = (xk)∞k=1 ∈ wp has

a unique representation ([14])

(5.3) x = ξ · e +

∞∑

(xk − ξ)e(k) where ξ is the strong limit of x;

wp∞ has no Schauder basis ([21, Remark 5.4 (a)]).

Notations for the duals of wp0, wp and w

We write

‖a‖Mp=

∞∑

ν=02ν max

2ν≤k≤2ν+1−1|ak| (p = 1)

∞∑

ν=02ν/p

2ν+1−1∑

k=2ν|ak|

(1 < p < ∞)

andMp =

a ∈ ω : ‖a‖Mp< ∞

The duals of wp0, wp and w

Theorem 5.2 Let † denote any of the symbols α, β and γ. Then

(a) (wp0)† = (wp)† = (w

p∞)† = Mp ([21, Theorem 5.5 (a)]);

(b) (wp0)∗ ≡ Mp ([21, Theorem 5.5 (b)]);

(c) ([14]) f ∈ (wp)∗ if and only if

f (x) = χf · ξ +

∞∑

akxk where a =(

f (e(k)))∞

k=1∈ Mp,

ξ is the strong limit of x, and

χf = f (e) −∞∑

f (e(k)), and ‖f‖ = |χf | + ‖a‖Mp;

(d) (wp∞)∗ is not given by any sequence space, but

‖ · ‖∗w

= ‖ · ‖Mpon (w

p∞)β ([21, Remark 5.6 and Theorem 5.5 (a)]).

The second duals of wp0, wp and w

Theorem 5.3 ([21, Theorem 5.7])The set Mp is a BK space with AK with ‖ · ‖Mp.

Theorem 5.4 (a) ([21, Theorem 5.8 (a)])Let † denote any of the symbols α, β and γ.

Then we have(w

p0)†† = (wp)†† = (w

p∞)†† = w

(b) ([21, Theorem 5.8 (b)])The continuous dual (Mp)

∗ of Mp is norm isomorphic with (wp∞, ‖ · ‖b).

Matrix transformations

Theorem 5.5 ([4, Theorem 2.4]) The necessary and sufficient conditionsfor A ∈ (X, Y ) when

X ∈ {wp0, w

p, wp∞} and Y ∈ {ℓ∞, c, c0}

can be read from the following table

FromTo

wp∞ w

ℓ∞ 1. 1. 1.c0 2. 3. 4.c 5. 6. 7.

1. (1.1) supn ‖An‖Mp< ∞

2. (2.1) limn→∞ ‖An‖Mp= 0

3. (1.1) and (3.1) limn→∞ ank = 0 for all k

4. (1.1), (3.1) and (4.1) limn→∞∑∞

k=1ank = 0

5. (5.1) αk = limn→∞ ank exists for all k,(5.2) (αk)∞k=1, An ∈ Mp for all n,(5.3) limn→∞ ‖An − (αk)∞k=1‖Mp

6. (1.1) and (5.1)

7. (1.1), (5.1) and (7.1) α = limn→∞∑∞

k=1ank exists.

Remark 5.6 The conditions for A ∈ (wp∞, c0) and A ∈ (w

p∞, c) can be

replaced by2’. (3.1) and (2.1’) ‖An‖Mp

converges uniformly in n

5’. (2.1’) and (5.1).

An estimate for ‖L‖χ when L ∈ B(wp0, c)

We obtain as an immediate consequence of Theorem 3.14:

Corollary 5.7 ([3, Corollary 3.6]) Let L ∈ B(wp0, c).

Then we have

2· limr→∞

supn≥r

‖An − α‖Mp

≤ ‖L‖χ ≤ limr→∞

supn≥r

‖An − α‖Mp

Representation of L ∈ B(X, c0) and equation for ‖L‖χ

Corollary 5.8 ([3, Corollary 3.7]) Let X be a BK space with AK.

Then every operator L ∈ B(X, c0) can be represented by an infinite com-plex matrix A ∈ (X, c0) such that L(x) = Ax for all x ∈ X ; also

‖L‖χ = limr→∞

supn≥r

‖An‖∗X

In particular, if L ∈ B(wp0, c0), then we have ([3, Corollary 3.8])

‖L‖χ = limr→∞

supn≥r

‖An‖Mp

Representation of L ∈ B(wp, c)

Theorem 5.9 ([3, Theorem 3.9]) (a) Every L ∈ B(wp, c) is given by amatrix B = (bnk)∞n=1,k=0 such that

L(x) = (bn0ξ +∑∞

k=1bnkxk)∞n=1 for all x ∈ wp,

where ξ ∈ C is the strong C1–limit of x,

(5.4) βk = limn→∞

bnk exists for all k ≥ 1, β = limn→∞

∞∑

bnk exists,

‖L‖ = supn∈N

|bn0| + ‖Bn‖Mp

Furthermore, we have

η = limn→∞

(L(x))n = ξβ +∑∞

k=1βk(xk − ξ)

β −∑∞

k=1βk

ξ +∑∞

k=1βkxk for all x ∈ wp.

Estimate for ‖L‖χ when L ∈ B(wp, c)

(b) If L ∈ B(wp, c), then we have

2· limn→∞

bn0 − β +

∞∑

∥(bnk − βk)∞k=1

≤ ‖L‖χ ≤ limn→∞

bn0 − β +

∞∑

∥(bnk − βk)∞k=1

where bnk (n ∈ N; k ∈ N0) are the entries of the matrix B that representsL by Part (a), and β and βk (k ∈ N0) are given by and (5.4).

We obtain the following corollaries from Theorem 5.9:

Corollary 5.10 ([3, Corollary 3.10]) Let A ∈ (wp, c). Then we have

n→∞

∑∞k=1αk − α

∣ + ‖An − (αk)∞k=1‖Mp

≤ ‖A‖χ

≤ limn→∞

∑∞k=1αk − α

∣ + ‖An − (αk)∞k=1‖Mp

where αk = limn→∞ ank for all k ∈ N and α = limn→∞∑∞

k=1 ank.

Corollary 5.11 ([3, Corollary 3.11]) Let L ∈ B(wp, c0). Then we have

‖L‖χ = limn→∞

|bn0| +∥

∥(bnk)∞k=1

with bnk = b(n)k (n ∈ N; k ∈ N0) from Theorem 5.9.

Corollary 5.12 ([3, Corollary 3.12]) Let A ∈ (wp, c0). Then we have

‖A‖χ = ‖LA‖χ = limn→∞

‖An‖Mp.

Characterisations of the compact operators L ∈ B(X, Y ) forX = w

p∞ and Y = c0, c

Corollary 5.13 ([3, Corollary 3.13]) Let L ∈ B(X, Y ). Then the necessaryand sufficient conditions for L ∈ C(X, Y ) can be read from the table

FromTo

wp0 wp

c0 1. 2.c 3. 4.

1. limr→∞

(supn≥r‖An‖Mp) = 0;

2. limn→∞

(|bn0 + ‖Bn‖Mp) = 0;

3. limr→∞

(supn≥r‖An − (αk)∞k=1‖Mp) = 0;

4. limn→∞

∣bn0 − β +∑∞

k=1βk

∣ + ‖Bn − (βk)∞k=1‖Mp

Characterisations of the compact matrix operators in(wp, c0) and (wp, c)

Corollary 5.14 ([3, Corollary 3.14])(a) Let A ∈ (wp, c0). Then we have LA ∈ C(wp, c0) if and only if

limn→∞

‖An‖Mp= 0.

(b) Let A ∈ (wp, c). Then we have LA ∈ C(wp, c) if and only if

limn→∞

∞∑

αk − α

+ ‖An − (αk)∞k=1‖Mp

with αk (n ∈ N) and α from Corollary 5.10.

Characterisation of the compact matrix operators in(w

p∞, c0) and (w

p∞, c)

Corollary 5.15 ([3, Corollary 3.15])(a) Let A ∈ (w

p∞, c0). Then we have LA ∈ C(w

p∞, c0) if and only if 1. in

Corollary 5.13 holds.

(b) Let A ∈ (wp∞, c). Then we have LA ∈ C(w

p∞, c0) if and only if 3. in

Corollary 5.13 holds.

Strong C1–regularity

We call an operator L ∈ B(wp, c) strongly C1–regular, if

limn→∞Ln(x) = ξ for all x ∈ wp,

where ξ is the strong C1–limit of x.A matrix A ∈ (wp, c) is said to be strongly C1–regular, if the operatorLA is strongly C1–regular.

Characterisation of compact strongly C1–regular operators

Corollary 5.16 ([3, Corollary 3.16]) Let L ∈ B(wp, c) be strongly C1–regular. Then we have L ∈ C(wp, c) if and only if

(5.5) limn→∞

|bn0 − 1| + ‖Bn‖Mp

with bnk = b(n)k (n ∈ N; k ∈ N0) from Theorem 3.16.

A strongly C1–regular matrix cannot be compact

Remark 5.17 ([3, Remark 3.17]) If A is a strongly C1–regular matrix thenLA cannot be compact, since we have with bn0 = 0 for all n ∈ N0 and

1 + ‖An‖Mp≥ 1 6= 0 for all n,

and so (5.5) in Corollary 5.16 cannot hold (Remark 3.20).

6 Graphical Representations of Neigh-bourhoods

We consider IRn for given n ∈ IN as a subset of ω by identifying every pointX = (x1, x2, · · · , xn) ∈ IRn with the real sequence x = (xk)∞k=1 ∈ ωwhere xk = 0 for all k > n, and introduce some of the metrics of theprevious sections on IRn.

LetBd(r, X0) = {X ∈ IRn : d(X, X0) < r} and ∂Bd(X0)

denote the open ball in (IRn, d) of radius r > 0 with its centre in X0, andits boundary.

We use the boundaries ∂Bd(X0) for the graphical representations of neigh-bourhoods.

∂Bd(p)(X0)

Example 6.1 We consider the metric d(p) of Example 2.4 (b).

Then ∂Bd(p)(r, X0) is given by a parametric representation

~x((u1, u2)) = (φ1((u1, u2)), φ2((u1, u2)) φ3((u1, u2))) + ~x0

for ((u1, u2) ∈ D = (−π/2, π/2) × (0, 2π))

with (Figure 6.1)

φ1((u1, u2))= rM/p1 · sgn(cos u2)(cos u1| cos u2|)2/p1,

φ2((u1, u2))= rM/p2 · sgn(sin u2)(cos u1| sin u2|)2/p2,

φ3((u1, u2))= φ3(u1) = rM/p3 · sgn(sin u1)| sin u1|2/p3.

Representation of ∂Bd(p)(X0) in Example 6.1

Figure 6.1 ∂Bd(p)(X0, r) for

p = (1/2, 2, 3/2); p = (1/2, 4, 1/4)

Neighbourhoods in a relative topology

Figure 6.2 Neighbourhoods in the relative topology on Enneper’s surfaceof the metrics d(p) of Figure 6.1

Neighbourhoods in a weak topology

Figure 6.3 Neighbourhoods on a sphere in the weak topology by the stere-ographic projection

7 Wulff’s Crystals and Potential Surfaces

Here we deal with Wulff’s construction, and the graphical representation ofWulff’s crystals and their surface energy functions as potential surfaces.

Wulff’s principle

According to Wulff’s principle ([28]), the shape of a crystal is uniquelydetermined by its surface energy function.A surface energy function is a real–valued function depending on adirection in space.

Let Bn and ∂Bn denote the unit sphere and its boundary in euclideanIRn+1.

Potential surfaces

Let F : ∂Bn → IR be a surface energy function. Then we may considerthe set

PM = {~x = F (~e)~e ∈ IRn+1 : ~e ∈ ∂Bn}

as a natural representation of the function F .

If n = 2, then

~e = ~e(u1, u2) = (cos u1 cos u2, cos u1 sin u2, sin u1)

for (u1, u2) ∈ R = (−π/2, π/2) × (0, 2π)

and we obtain a potential surface with a parametric representation(Figure 7.1)

PS = {~x = f (u1, u2)(cos u1 cos u2, cos u1 sin u2, sin u1) : (u1, u2) ∈ R}

where f (u1, u2) = F (~e(u1, u2)).

Figure 7.1 A potential surface

Representations of potential surfaces and correspondingWulff’s crystals

Figure 7.2 Potential surfaces and corresponding Wulff’s crystals

Representations of potential surfaces and correspondingWulff’s crystals

Figure 7.3 Potential surfaces and corresponding Wulff’s crystals

Wulff’s principle

Let • denote the usual inner product in IRn.

Wulff gave a geometric principle of construction for crystals.

Theorem 7.1 (Wulff’s principle) ([8, p. 88]) For every ~e ∈ ∂Bn, letE~e denote the hyperplane orthogonal to ~e and through the point P withposition vector ~p = F (~e)~e, and H~e be the half space which contains theorigin and has the boundary E~e = ∂H~e. Then the crystal CF which hasF as its surface energy function is uniquely determined and given by

CF =⋂

~e∈∂Bn

H~e =⋂

~e∈∂Bn

{~x : ~x • ~e ≤ F (~e)}.

Remark 7.2 It is clear that if the surface energy function F is continuousthen CF is a closed convex subset of IRn+1.

Theorems on Wulff’s construction

Theorem 7.3 ([8, Satz 6.1]) Let F : ∂Bn →IR+ be a continuous function.Then a point X with position vector ~x is on the boundary ∂CF of Wulff’scrystal CF corresponding to F if and only if (left in Figure 7.4)

F (~e) ≥ ~x • ~e for all ~e ∈ ∂Bn and F (~e0) = ~x • ~e0 for some ~e0 ∈ ∂Bn.

Theorem 7.4 ([8, Satz 6.2]) Let F : ∂Bn →IR+ be a continuous functionand CF : ∂Bn →IR+ be defined by

(7.6) CF (~e) = inf{

F (~u)(~e • ~u)−1 : ~u ∈ ∂Bn and ~e • ~u > 0}

Then the boundary ∂CF of Wulff’s crystal corresponding to F is given by

(7.7) ∂CF = {~x = CF (~e)~e ∈ IRn+1 : ~e ∈ ∂Bn}.

Wulff’s crystal for n = 2

Example 7.5 If n = 2, then a parametric representation for the boundary∂CF of Wulff’s crystal corresponding to F is (right in Figure 7.4)

(7.8) ~x(u1, u2) = CF (~e(u1, u2))~e(u1, u2)

for (u1, u2) ∈ R = (−π/2, π/2) × (0, 2π).

Wulff’s constructions according to Theorems 7.1 and 7.3

Figure 7.4 Wulff’s constructions according to Theorems 7.1 and 7.3

Potential surfaces and Wulff’s crystals constructed byTheorems 7.1 and 7.3

Figure 7.5 Wulff’s crystals constructed by Theorems 7.3 and 7.4

The special case F = ‖ · ‖

When F is equal to a norm ‖ · ‖ in IR3, then the boundary of the corre-sponding Wulff’s crystal is given by the dual norm of ‖ · ‖.

Corollary 7.6 Let ‖ · ‖ be a norm on IRn+1 and, for each ~w ∈ ∂Bn, letφ~w :IRn+1 → IR be defined by

φ~w(x) = ~w • ~x =

n+1∑

wkxk (~x ∈ IRn+1).

Then the boundary ∂C‖·‖ of the corresponding Wulff’s crystal is given by

(7.9) ∂C‖·‖ =

‖φe‖∗· ~e ∈ IRn+1 : ~e ∈ ∂Bn

where ‖φ~e‖∗ is the norm of the functional φ~e, hence the dual norm of ‖·‖.

The dual cases F = ‖ · ‖1 and F = ‖ · ‖∞

Figure 7.6 Wulff’s crystals corresponding to the norms ‖ · ‖1 and ‖ · ‖∞

The case F = ‖ · ‖wp∞

Figure 7.7 Potential surface of the wp∞ norm and potential surface with

corresponding Wulff’s crystal

The case F = ‖ · ‖Mp

Figure 7.8 Potential surface of the Mp norm and potential surface withcorresponding Wulff’s crystal

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