Summability, Matrix Transformations and Their Applications Doctoral School, Tartu University, Tartu, Estonia 13 October, 2011

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1 Summability, Matrix Transformations and Their Applications Doctoral School, Tartu University, Tartu, Estonia 13 October, 2011 Eberhard Malkowsky Department of Mathematics Faculty of Arts and Sciences Fatih University Büyükçekmece, Istanbul Turkey 1 Research supported by the research project # 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 1

2 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 c 0 5 The Spaces of Strongly Summable and Bounded Sequences 2

3 6 Graphical Representations of Neighbourhoods 7 Wulff s Crystals and Potential Surfaces References 3

4 1 Introduction The theory of matrix transformations is a wide field in summability; it deals with the characterisations of classes of matrix mappings between sequence spaces by giving necessary and sufficient conditions on the entries of the infinite 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 systems of linear equations 4

5 Summability The classical summability theory deals with a generalization of convergence of sequences and series. One original idea was to assign a limit to divergent sequences or series. Classical methods of summability were applied to problems in analysis such as the analytic continuation of power series and improvement of the rate of 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 Cesàro matrices C α Euler matrices E q Hölder matrices H α Nörlund matrices (N, p) 5

6 Concepts of summability Let A = (a nk ) n, be an infinite matrix, x = (x k) be a sequence, e = (1, 1, 1 ), A n x = a nk x k and Ax = (A n x) n=1 be the sequence of the A transforms of x. There are three concepts of summability. Ordinary summability: x is summable A if lim n A nx = l for some l C strong summability: x is strongly summable A with index p > 0 if lim n A n( x l e p ) = lim a n nk x k l p = 0 for some l C absolute summability: x is absolutely summable A with index p > 0 if A n x A n 1 x p <. n=1 6

7 An example Example 1.1 Let the matrix A be given by a nk = 1/n for 1 k n and a nk = 0 for k > n (n = 1, 2,... ). Then the A transforms of the sequence x are the arithmetic means of the terms of x, that is, σ n = 1 n x n k and A defines the Cesàro method C 1 of order 1. Every convergent sequence is summable C 1 and the limit is preserved the divergent sequence (( 1) k ) is summable C 1 to 0 strong summability of index 1 implies ordinary summability to the same limit; the converse is not true, in general absolute summability with index 1 implies ordinary summability. 7

8 2 Matrix Transformations Let X and Y be subsets of the set ω of all complex sequences. The theory of 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) A n x converges for all n and all x X and (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 8

9 2.1 Notations bounded l c c 0 φ the sets of all convergent null finite sequences x = (x k ) e e (n) (n IN) the sequences with e k = 1, (k = 1, 2,... ) e (n) k = { 1 (k = n) 0 (k n) 9

10 B X (r, x 0 ), S X (r, x 0 ) and X Let (X, d) be a metric space. Then B X (r, x 0 ) = {x X : d(x, x 0 ) < r} is the open ball and S X (r, x 0 ) = {x X : d(x, x 0 ) = r} is the sphere in X of radius r > 0 and centre in x 0 X; we write B X = B X (1, 0) and S X = S X (1, 0) if X is a linear metric space then X is the continous dual of X, that is, 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 = sup x SX L(x) X = B(X, C) is the continuous dual of X with f = sup x SX f(x) for all f X 10

11 2.2 FK, BK and AK spaces The theory of FK, BK and AK spaces from modern functional analysis is the most powerful tool in the characterisation of matrix transformations. The reason is that matrix transformations between FK spaces are continuous. The theory, however, fails in some cases, for instance, when the initial sequence 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 is applied. 11

12 F K and BK spaces ([27, p. 55] A subspace X of ω is said to be an FK space if it is a Fréchet space, that is, a complete, locally convex, linear metric space, with continuous coordinates P n : X C (n = 1, 2,... ) where P n (x) = x n ; a BK space is an FK space with its metric given by a norm. AK property [27, Definition ] An FK space X φ is said to have AK if every sequence x = (x k ) X has a unique representation n x = x k e (k), that is, x = lim x n k e (k). 12

13 (ω, d ω ) Example 2.2 The set ω is a Fréchet space with ([26, Example ]) 1 (2.3) d ω (x, y) = 2 k x k y k for all x, y ω 1 + x k y k and (i) convergence in (ω, d ω ) is equivalent to coordinatewise convergence ([26, Theorem and Example 4.1.5]). So (ω, d ω ) is an FK space, which obviously has AK. Remark 2.3 In view of (i), an FK space X is a (ii) Fréchet sequence space with its metric stronger than the metric d ω of ω on X, or equivalently, (iii) a Fréchet sequence space which is continuously embedded in ω. 13

14 The FK spaces l(p) and c 0 (p) Example 2.4 Let p = (p k ) be a bounded sequence of positive reals with M = max{1, sup k p k }. Then l(p) = {x ω : x k p k < } and c 0 (p) = {x ω : lim x k p k = 0} k are FK spaces with AK with 1/M d (p) (x, y) = x k y k p k ([15, Theorem 1]) and d 0,(p) (x, y) = sup k x k y k p k/m ([18, Theorem 2]). 14

15 The classical BK spaces Example 2.5 (a) The sets l p (1 p < ), c 0, c and l are BK spaces with 1/p x p = x k p and x = sup x k ([26, Example ]); k c 0 is a closed subspace of c and c is a closed subspace of l ([27, Corollary 4.2.4]); l p and c 0 have AK ([27, Example 7.3.7]); every sequence x = (x k ) c has a unique representation x = ξ e + (x k ξ)e (k) where ξ = lim x k ([17, Example ]); k l has no Schauder basis, since it is not separable ([17, Theorem 3.4.7, Problem 3.4.4]). 15

16 2.3 The dual spaces If X, Y ω, then M(X, Y ) = { a ω : ax = (a k x k ) Y for all x X} is called multiplier space of X and Y; the sets X α = M(X, l 1 ), X β = M(X, cs) and X γ = M(X, cs) are called the α, β and γ duals of X. Properties of duals of BK spaces Theorem 2.6 ([27, Theorem ]) Let X and Y be BK spaces. Then M(X, Y ) is a BK space with z = sup{ xz : x = 1} (z M(X, Y )); in particular, the X α, X β, and X γ are BK spaces. 16

17 Theorem 2.6 does not extend to FK spaces, in general Example 2.7 The space (ω, d ω ) is an FK space (Example 2.2) and ω β = φ, as we will see in Example 2.10 (i), but φ has no Fréchet topology ([27, 4.0.5]). Relation between X and X β Theorem 2.8 (a) If X φ is an FK space with AK then X β = X γ ([27, Theorem (iii)]). (b) ([27, Theorem 7.2.9]) Let X φ be an FK space. Then X β X in the sense that each sequence a X β can be used to represent a function f a X with f a (x) = a k x k for all x X, and the map T : X β X with T(a) = f a is linear and one to one. If X has AK then T is an isomorphism. 17

18 The norm X Let X ω be a normed sequence space, and a ω. Then we write a X = sup a k x k x S X provided the expression on the righthand side exists and is finite which is the case whenever X is a BK space and a X β by Theorem 2.8 (b). Norm isomorphic, If X and Y are norm isomorphic then we write X Y. 18

19 The β duals of l(p) and c 0 (p) Example 2.9 (a) If p k > 1 and q k = p k /(p k 1) for all k, then (l(p)) β = a ω : a k q k < N N>1 ([16, Theorem 11]). (b) If p is any sequence of positive reals then, (c 0 (p)) β = a ω : a k N 1/p k < N>1 ([16, Theorem 6]). 19

20 The duals of the classical BK spaces Example 2.10 ([26, Examples 6.4.2, and 6.4.4]) We have (i) ω β = φ and φ β = ω (ii) l 1 lβ 1 = l ; l p l β p = l q for 1 < p < and q = p/(p 1) (iii) c 0 cβ 0 = cβ = l β = l 1 (iv) f c if and only if f(x) = χ f lim x k + a k x k where a = (f(e (k) )) k l 1 and χ f = f(e) f(e (k) ), and f = χ f + a 1 (v) l is not given by any sequence space ([26, Example 6.4.8]), but (2.4) l = 1 on l β. 20

21 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 ), then L A B(X, Y ) where L A (x) = Ax (x X) ([27, Theorem 4.2.8]). (b) If X has AK then we have B(X, Y ) (X, Y ); this means every L B(X, Y ) is given by a matrix A (X, Y ) such that L(x) = Ax (x X) ([12, Theorem 1.9]). 21

22 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 ) = sup n A n X < ; moreover, if A (X, Y ) then L A = A (X,l ). Improvement of mapping Theorem 2.13 (a) ([27, 8.3.6]) Let X be an FK space with φ = X, Y and Y 1 be FK spaces with Y 1 a closed subspace of Y. Then we have A (X, Y 1 ) if and only if A (X, Y ) and Ae (k) Y 1 for all k. (b) ([27, 8.3.7]) Let X be an FK space and X 1 = X e. Then A (X 1, Y ) if and only if A (X, Y ) and Ae Y. 22

23 Mapping properties of the transpose Let A t denote the transpose of the matrix A. Theorem 2.14 ([27, Theorem 8.3.9]) Let X and Z be BK spaces with AK and Y = Z β. Then we have (a) (X ββ, Y ) = (X, Y ); (b) A (X, Y ) if and only if A t (Z, X β ). 23

24 The classes (X, Y ) for X = l, c, c 0 Example 2.15 We have (a) A (l, l ) if and only if (2.5) A (, ) = sup n if A (l, l ) then a nk < ([27, 8.4.5A]); (2.6) L A = A (, ) (b) (c 0, l ) = (c, l ) = (l, l ) ([27, 8.4.5A]). Proof. (a) Since l is a BK space by Example 2.5 (a), the statements follow from Theorem 2.12 and (2.4). (b) Since c 0 is a BK space with AK by Example 2.5 (a), and c ββ 0 = l β 1 = l by Example 2.10 (iii) and (ii), Theorem 2.14 (a) yields (c 0, l ) = (l, l ); this and c 0 c l yield (c, l ) = (l, l ). 24

25 The class B(l 1, l 1 ) Example 2.16 ([20, Theorem 2.27]) We have B(l 1, l 1 ) = (l 1, l 1 ) by Theorem 2.11 and Example 2.5 (b). Also A (l 1, l 1 ) if and only if (2.7) A (1,1) = sup a nk < k n=1 moreover, if L B(l 1, l 1 ) then (2.8) L = A (1,1). Proof. We apply Theorem 2.14 (b) with X = l 1 and Z = c 0, BK spaces with AK by Example 2.5, and Y = c β 0 = l 1 by Example 2.10 (iii) to obtain A (l 1, l 1 ) if and only if A t (l, l ), and this is the case by (2.5) if and only if A t (, ) = sup n a kn = A (1,1) <. 25

26 Furthermore, if L (l 1, l 1 ), then L(x) 1 = Ax 1 = A n x = a nk x k n=1 n=1 x k a nk A (1,1) x 1 n=1 implies (2.9) L A (1,1). We also have for e (k) S l1 (k IN) L(e (k) ) 1 = Ae (k) 1 = A n e (k) = n=1 hence A (1,1) L. This and (2.9) imply (2.8). n=1 a nk L, 26

27 Regular matrices, the Silverman Toeplitz theorem A matrix A (c, c) is said to be regular if lim n A n x = lim k x k for 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 (, ) = sup a nk < ((2.5)), n (ii) α k = lim n a nk exists for each k IN, (iii) α = lim a n nk exists. 27

28 (b) Let A (c, c) and x c. Then we have (2.10) lim n A nx = α α k lim x k + k α k x k. (c) A matrix A is regular if and only if (i) holds and (ii ) lim n a nk = 0 for each k IN, (iii ) lim a n nk = 1. Proof. Part (a) follows from Theorem 2.13 (a) and (b) and Example (b) This is elementary. Part (c) follows from Parts (a) and (b). 28

29 Schur s theorem; proof by the method of the gliding hump Theorem 2.18 (Schur) We have (a) ([27, Theorem ]) A (l, c) if and only if (2.11) a nk converges uniformly in n and (2.12) lim n a nk = α k exists for each k IN; (b) ([27, Theorem ]) A (l, c 0 ) if and only if condition (2.11) holds and (2.13) lim n a nk = 0 for each k IN. Remark 2.19 The conditions in (2.11) and (2.13) are equivalent to a nk = 0 ([25, 21. (21.1)]). lim n 29

30 Applications Example 2.20 (Steinhaus) A regular matrix cannot sum all bounded sequences. Proof. If there were a matrix regular A (l, c), then it would follow from the conditions in (2.11) of Theorem 2.18 and in (ii ) and (iii ) in Part (c) of Theorem 2.17 that 1 = lim n a nk = lim n a nk = 0, a contradiction. Example 2.21 Schur s theorem can be applied to show that weak and strong convergence coincide in l 1. 30

31 Proof of Example 2.21 We assume that the sequence (x (n) ) n=1 is weakly convergent to x in l 1, 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) = a k y k for all y l 1. We define the matrix B = (b nk ) n, by b nk = x (n) k x k (n, k = 1, 2,... ). Then we have for all a l f(x (n) ) f(x) = f(x (n) x) = a k (x (n) k x k ) 31

32 = b nk a k 0 (n ), that is, B (l, c 0 ). It follows from Theorem 2.18 (b), that b nk converges uniformly in n and lim n b nk = 0 for each k. Thus we have x (n) x 1 = x (n) k x k = b nk 0 (n ). 32

33 3 Compact Operators The Hausdorff measure of noncompactness is most effective in determining conditions for a linear operator to be compact. Measures of noncompactness are also very useful tools in the theory of operator equations in Banach spaces, in particular in fixed point theory. The first measure of noncompactness, denoted by α, was defined and studied by Kuratowski [13] in Later, in 1955, Darbo [6] used the function α to prove a generalisation of Schauder s fixed point theorem to noncompact operators. Other measures of noncompactness, the Hausdorff measure of noncompactness, were introduced by Goldenštein, Gohberg and Markus [9] in 1957, and later studied by Goldenštein and Markus [10] in 1968, and by Istrǎţesku [11] in The general theory of measures of noncompactness can be found in [1, 2]. 33

34 Notations We write M X 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 D L = X for the domain D L of L for every bounded sequence (x n ) in X, the sequence (L(x n )) has a convergent 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 x k x k for all k implies x x. 34

35 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 : M X [0, ) is said to be a measure of noncompactness on X (MNC) if it satisfies the 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 M X (Invariance under closure) (MNC.3) µ(q 1 Q 2 ) = max{µ(q 1 ), µ(q 2 )} for all Q 1, Q 2 M X (Semi additivity) 35

36 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, Q 1, Q 2 M X (MNC.1 ) Q 1 Q 2 implies µ(q 1 ) µ(q 2 ) (Monotonicity) (MNC.2 ) µ(q 1 Q 2 ) min{µ(q 1 ), µ(q 2 )} (MNC.3 ) µ(q) = 0 for every finite set Q (Non singularity) 36

37 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, Q 1, Q 2 M X and all scalars λ (MNC.4 ) µ(q 1 + Q 2 ) µ(q 1 ) + µ(q 2 ) (Algebraic semi additivity) (MNC.5 ) (MNC.6 ) µ(λq 1 ) = λ µ(q 1 ) (Homogenity) µ(x + Q) = µ(q) (Translation invariance) 37

38 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 M X. The Hausdorff measure of noncompactness (HMNC) of Q is defined by n χ(q) = inf ǫ > 0 : Q B(r i, x i ); r i < ǫ, x i X. i=1 38

39 Invariance of the HMNC under the passage to the convex hull 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 M X 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(b X ) = 1. 39

40 HMNC of bounded sets in BK spaces Theorem 3.7 ([3, Theorem 3.4]) (a) Let X be a monotone BK space with AK and P n : 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) = lim n sup (I P n )(x) x Q for all Q M X. (b) Let P n : c c be the projectors onto the linear span of {e, e (1),..., e (n) } for n = 1, 2,.... Then ( ) (3.3) 1 2 lim n sup (I P n )(x) x Q ( lim n χ(q) sup (I P n )(x) x Q ) for all Q M X. 40

41 HMNC of operators between Banach spaces Definition 3.8 ([20, Definition 2.24]) Let X and Y be Banach spaces and χ 1 and χ 2 be HMNC s on X and Y. Then the operator L : X Y is called (χ 1, χ 2 )-bounded if L(Q) M Y for every Q M X there exists a constant C > 0 such that (3.4) χ 2 (L(Q)) C χ 1 (Q) for every Q M X. If an operator L is (χ 1, χ 2 )- bounded then the number L (χ1,χ 2 ) = inf{c > 0 : (3.4) holds} is called the (χ 1, χ 2 )- measure of noncompactness of L. If χ 1 = χ 2 = χ, then we write L χ = L (χ,χ). 41

42 Properties of the HMNC of operators Theorem 3.9 ([20, Theorem 2.25]) Let X and Y be Banach spaces and L B(X, Y ). Then (3.5) L χ = χ(l(s X )) = χ(l(b X )). 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 (3.6) L χ = 0 if and only if L C(X, Y ), L χ L, L + K χ = L χ for each K C(X, Y ), L L χ L χ L χ. 42

43 L χ for L B(l 1, l 1 ) Example 3.11 (Goldenštein, Gohberg and Markus) ([20, Theorem 2.28]) Let L B(l 1, l 1 ). Then we have by Example 2.16, (2.7), (2.8), (3.2) and (3.5) ( ) (3.7) L χ = L A χ = lim m sup k n=m a nk. Example 3.12 (Goldenštein, Gohberg and Markus) ([20, Corollary 2.29]) Let L B(l 1, l 1 ). Then we have by (3.6) and (3.7), L C(l 1, l 1 ) if and only if ( ) (3.8) lim m sup k n=m a nk = 0. 43

44 Compact matrix operators between the classical sequence spaces Remark 3.13 (a) It is clear from (3.8) that I B(l 1, l 1 ) \ C(l 1, l 1 ). (b) The necessary and sufficient conditions for an operator given by a matrix A (X, Y ) to be compact can be found in [23, (a) (b), p. 85] when X = l 1, l, c 0, c, Y = l r for 1 r < ; X = l p for 1 < p, Y = l 1, l ; X = c 0, Y = l. (c) The class C(l 1, l ) cannot be determined by the use of the Hausdorff measure of noncompactness. The characterisation of the class C(l 1, l ) is established in [23, Theorem 5]. 44

45 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 = (a nk ) n, (X, c) such that L(x) = Ax for all x X (Theorem 2.11 (b)). The Hausdorff measure of noncompactness of L satisfies ( ) 1 (3.9) 2 r lim A n α X L χ where sup n r lim r ( sup n r A n α X (3.10) α k = lim k a nk for every k IN and (α k ) Xβ. ), 45

46 Representation of L B(c, c) Lemma 3.15 Every L B(c, c) is given by a matrix B = (b nk ) n=1,k=0 such that L(x) = (b n0 ξ + b nk x k ) n=1 (3.11) β k = lim n b nk exists for all k 1, Furthermore, we have L = sup n N for all x c, where ξ = lim k x k, b nk. k=0 β = lim n b nk exists, k=0 η = lim n (L(x)) n = ( β β k ) ξ + β k x k for all x c. 46

47 A formula for L χ when L B(c, c) Theorem 3.16 ([19, Theorem 1]) If L B(c, c), then we have 1 2 n lim b n0 β + β k + b nk β k L χ lim n b n0 β + β k + b nk β k, where b nk (n IN;k IN 0 ) are the entries of the matrix B that represents L by Lemma 3.15, and β and β k (k IN) are given by (3.11). 47

48 Characterisation of C(X, Y ) when X, Y = c 0, c Corollary 3.17 Let L B(X, Y ). Then the necessary and sufficient conditions for L C(X, Y ) can be read from the table 1. lim r (sup n r From c 0 c To c c a nk ) = 0; where 2. lim n ( k=0 b nk ) = 0; 3. lim r (sup n r a nk α k ) = 0; ( 4. lim bn0 β + n β k + b nk β k ) = 0. 48

49 Compact matrix operators in (c, c 0 ) and (c, c) Corollary 3.18 (a) Let A (c, c 0 ). Then we have L A C(c, c 0 ) if and only if a nk = 0. lim n (b) Let A (c, c). Then we have L A C(c, c) if and only if lim n α k α + a nk α k = 0 with α k (k IN) from (3.10) or (ii) in Theorem 2.17, and α = lim a n nk ((iii) in Theorem 2.17). 49

50 Regular operators An operator L B(c, c) is said to be regular, if lim n L n (x) = ξ for all x c, where ξ = lim k x k. Corollary 3.19 ([5, Corollary 3]) Let L B(c, c) be regular. Then we have L C(c, c) if and only if (3.12) lim n ( b n0 1 + b nk ) = 0 with b nk (n IN;k IN 0 ) from Theorem Remark 3.20 If A is a regular matrix then L A cannot be compact, since we have b n0 = 0 for all n IN and 1 + a nk 1 0 for all n, and so (3.12) in Corollary 3.19 cannot hold. 50

51 4 Fredholm Operators on c 0 Here we apply our previous results to establish sufficient conditions for a matrix operator in c 0 to be a Fredholm operator. The presented results are the special cases T = I of those in [7] If we denote the set of finite rank operators by F(X, Y ), and suppose that X is a normed space and Y is a Banach space, then it is well known that F(X, Y ) C(X, Y ) ([22], p.111). In particular, if X is a Banach space and Y is a Hilbert space, then the set of compact operators is the closure of the 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 Fredholm operators. Here, we will use a result based on the relation between them and compact operators. 51

52 Fredholm operators Definition 4.1 Let X and Y be Banach spaces and L B(X, Y ). We denote the null and the range spaces of L by N(L) and R(L). Then L is said 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 ); we write 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. 52

53 A sufficient condition for B(X) Φ(X) Theorem 4.4 Let L B(c 0 ). If (4.1) lim r sup n r then we have L Φ(c 0 ). 1 a nn +,k n a nk = 0, Proof. We put C = I A where A is the matrix with Ax = L(x) for all x c 0 (Theorem 2.11 (b)). Then L C C(c 0, c 0 ) by 1. in Corollary 3.17 if and only if lim r sup n r c nk = 0 which is (4.1). Now the statement follows with Theorem

54 5 The Spaces of Strongly Summable and Bounded Sequences Let 1 p < throughout w p 0 = x ω : n lim 1 n w p = w p = x ω : lim n x ω : sup n 1 n 1 n n x k p = 0, n x k ξ p = 0 for some ξ C, n x k p <. 54

55 Strong limit If x w p then the strong C 1 limit of x is the number ξ C with (5.1) lim 1 n x n n k ξ p = 0. Sectional and block norms We define the sectional s and block norms b by 1/p (5.2) x s = sup 1 n x n n k p and x b = sup 1 ν 2 ν 2 ν+1 1 x k p k=2 ν 1/p. 55

56 Topological properties of w p 0, wp and w p Theorem 5.1 (a) The strong limit is unique for each x w p ([14]). (b) We have w p 0 wp w p ; s and b are equivalent norms on w p 0, wp and w p ([21, Proposition 5.3]); (c) w p 0, wp and w p are BK spaces with b ([21, Proposition 5.3]); w p 0 is a closed subspace of wp, and w p is a closed subspace of w p (Part (b) and [27, Corollary 4.2.4]); w p 0 has AK ([21, Remark 5.4 (b)]); every sequence x = (x k) wp has a unique representation ([14]) (5.3) x = ξ e + (x k ξ)e (k) where ξ is the strong limit of x; w p has no Schauder basis ([21, Remark 5.4 (a)]). 56

57 Notations for the duals of w p 0, wp and w p We write a Mp = ν=0 ν=0 2 ν max a k (p = 1) 2 ν k 2 ν+1 1 ( ) 2 ν+1 1/q 2 ν/p 1 a k q (1 < p < ). k=2 ν and M p = { } a ω : a Mp <. 57

58 The duals of w p 0, wp and w p Theorem 5.2 Let denote any of the symbols α, β and γ. Then (a) (w p 0 ) = (w p ) = (w p ) = M p ([21, Theorem 5.5 (a)]); (b) (w p 0 ) M p ([21, Theorem 5.5 (b)]); (c) ([14]) f (w p ) if and only if f(x) = χ f ξ + a k x k where a = ( ) f(e (k) ) M p, ξ is the strong limit of x, and χ f = f(e) f(e (k) ), and f = χ f + a Mp ; (d) (w p ) is not given by any sequence space, but w p = M p on (w p ) β ([21, Remark 5.6 and Theorem 5.5 (a)]). 58

59 The second duals of w p 0, wp and w p Theorem 5.3 ([21, Theorem 5.7]) The set Mp is a BK space with AK with M p. Theorem 5.4 (a) ([21, Theorem 5.8 (a)]) Let denote any of the symbols α, β and γ. Then we have (w p 0 ) = (w p ) = (w p ) = w p. (b) ([21, Theorem 5.8 (b)]) The continuous dual (M p ) of M p is norm isomorphic with (w p, b ). 59

60 Matrix transformations Theorem 5.5 ([4, Theorem 2.4]) The necessary and sufficient conditions for A (X, Y ) when X {w p 0, wp, w p } and Y {l, c, c 0 } can be read from the following table where From To w p w p 0 wp l c c

61 1. (1.1) sup n A n Mp < 2. (2.1) lim n A n Mp = 0 3. (1.1) and (3.1) lim n a nk = 0 for all k 4. (1.1), (3.1) and (4.1) lim n a nk = 0 5. (5.1) α k = lim n a nk exists for all k, (5.2) (α k ), A n M p for all n, (5.3) lim n A n (α k ) M p = 0 6. (1.1) and (5.1) 7. (1.1), (5.1) and (7.1) α = lim n a nk exists. Remark 5.6 The conditions for A (w p, c 0 ) and A (w p, c) can be replaced by 2. (3.1) and (2.1 ) A n Mp converges uniformly in n 5. (2.1 ) and (5.1). 61

62 An estimate for L χ when L B(w p 0, c) We obtain as an immediate consequence of Theorem 3.14: Corollary 5.7 ([3, Corollary 3.6]) Let L B(w p 0, c). Then we have ( 1 2 r lim sup n r A n α Mp ) L χ lim r ( sup n r A n α Mp ). 62

63 Representation of L B(X, c 0 ) 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, c 0 ) can be represented by an infinite complex matrix A (X, c 0 ) such that L(x) = Ax for all x X; also ( ) L χ = lim r sup n r A n X In particular, if L B(w p 0, c 0), then we have ([3, Corollary 3.8]) ( L χ = lim r sup n r. A n Mp ). 63

64 Representation of L B(w p, c) Theorem 5.9 ([3, Theorem 3.9]) (a) Every L B(w p, c) is given by a matrix B = (b nk ) n=1,k=0 such that L(x) = (b n0 ξ + b nk x k ) n=1 for all x wp, where ξ C is the strong C 1 limit of x, (5.4) β k = lim n b nk exists for all k 1, Furthermore, we have L = sup n N β = lim n ( b n0 + B n Mp ). η = lim n (L(x)) n = ξβ + β k (x k ξ) b nk exists, k=0 = ( β β k ) ξ + β k x k for all x w p. 64

65 Estimate for L χ when L B(w p, c) (b) If L B(w p, c), then we have 1 2 n lim b n0 β + β k + (b nk β k ) Mp L χ lim n b n0 β + β k + (bnk β k ) Mp, where b nk (n N;k N 0 ) are the entries of the matrix B that represents L by Part (a), and β and β k (k N 0 ) are given by and (5.4). We obtain the following corollaries from Theorem 5.9: 65

66 Corollary 5.10 ([3, Corollary 3.10]) Let A (w p, c). Then we have 1 ( 2 n lim α k α ) + A n (α k ) M p A χ ( lim α n k α ) + A n (α k ) M p where α k = lim n a nk for all k N and α = lim n a nk., Corollary 5.11 ([3, Corollary 3.11]) Let L B(w p, c 0 ). Then we have ( L χ = lim b n n0 + (b nk ) ) Mp with b nk = b (n) k (n N;k N 0 ) from Theorem 5.9. Corollary 5.12 ([3, Corollary 3.12]) Let A (w p, c 0 ). Then we have A χ = L A χ = lim n A n Mp. 66

67 Characterisations of the compact operators L B(X, Y ) for X = w p 0, wp and Y = c 0, c Corollary 5.13 ([3, Corollary 3.13]) Let L B(X, Y ). Then the necessary and sufficient conditions for L C(X, Y ) can be read from the table From To w p 0 wp c c where 1. lim r (sup n r A n Mp ) = 0; 2. lim n ( b n0 + B n Mp ) = 0; 3. lim r (sup n r A n (α k ) M p ) = 0; ( bn0 4. lim β + ) n β k + Bn (β k ) M p = 0. 67

68 Characterisations of the compact matrix operators in (w p, c 0 ) and (w p, c) Corollary 5.14 ([3, Corollary 3.14]) (a) Let A (w p, c 0 ). Then we have L A C(w p, c 0 ) if and only if lim n A n Mp = 0. (b) Let A (w p, c). Then we have L A C(w p, c) if and only if lim n α k α + A n (α k ) M p = 0 with α k (n N) and α from Corollary

69 Characterisation of the compact matrix operators in (w p, c 0 ) and (w p, c) Corollary 5.15 ([3, Corollary 3.15]) (a) Let A (w p, c 0 ). Then we have L A C(w p, c 0 ) if and only if 1. in Corollary 5.13 holds. (b) Let A (w p, c). Then we have L A C(w p, c 0 ) if and only if 3. in Corollary 5.13 holds. Strong C 1 regularity We call an operator L B(w p, c) strongly C 1 regular, if lim n L n (x) = ξ for all x w p, where ξ is the strong C 1 limit of x. A matrix A (w p, c) is said to be strongly C 1 regular, if the operator L A is strongly C 1 regular. 69

70 Characterisation of compact strongly C 1 regular operators Corollary 5.16 ([3, Corollary 3.16]) Let L B(w p, c) be strongly C 1 regular. Then we have L C(w p, c) if and only if ) (5.5) lim ( b n n0 1 + B n Mp = 0 with b nk = b (n) k (n N;k N 0 ) from Theorem A strongly C 1 regular matrix cannot be compact Remark 5.17 ([3, Remark 3.17]) If A is a strongly C 1 regular matrix then L A cannot be compact, since we have with b n0 = 0 for all n N 0 and 1 + A n Mp 1 0 for all n, and so (5.5) in Corollary 5.16 cannot hold (Remark 3.20). 70

71 6 Graphical Representations of Neighbourhoods We consider IR n for given n IN as a subset of ω by identifying every point X = (x 1, x 2,, x n ) IR n with the real sequence x = (x k ) ω where x k = 0 for all k > n, and introduce some of the metrics of the previous sections on IR n. Let B d (r, X 0 ) = {X IR n : d(x, X 0 ) < r} and B d (X 0 ) denote the open ball in (IR n, d) of radius r > 0 with its centre in X 0, and its boundary. We use the boundaries B d (X 0 ) for the graphical representations of neighbourhoods. 71

72 B d(p) (X 0 ) Example 6.1 We consider the metric d (p) of Example 2.4 (b). Then B d(p) (r, X 0 ) is given by a parametric representation x((u 1, u 2 )) = (φ 1 ((u 1, u 2 )), φ 2 ((u 1, u 2 )) φ 3 ((u 1, u 2 ))) + x 0 for ((u 1, u 2 ) D = ( π/2, π/2) (0, 2π)) with (Figure 6.1) φ 1 ((u 1, u 2 ))= r M/p1 sgn(cos u 2 )(cos u 1 cos u 2 ) 2/p 1, φ 2 ((u 1, u 2 ))= r M/p2 sgn(sin u 2 )(cos u 1 sin u 2 ) 2/p 2, φ 3 ((u 1, u 2 ))= φ 3 (u 1 ) = r M/p3 sgn(sinu 1 ) sin u 1 2/p 3. 72

73 Representation of B d(p) (X 0 ) in Example 6.1 Figure 6.1 B d(p) (X 0, r) for p = (1/2, 2, 3/2); p = (1/2, 4, 1/4) 73

74 Neighbourhoods in a relative topology Figure 6.2 Neighbourhoods in the relative topology on Enneper s surface of the metrics d (p) of Figure

75 Neighbourhoods in a weak topology Figure 6.3 Neighbourhoods on a sphere in the weak topology by the stereographic projection 75

76 7 Wulff s Crystals and Potential Surfaces Here we deal with Wulff s construction, and the graphical representation of Wulff 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 uniquely determined by its surface energy function. A surface energy function is a real valued function depending on a direction in space. Let B n and B n denote the unit sphere and its boundary in euclidean IR n+1. 76

77 Potential surfaces Let F : B n IR be a surface energy function. Then we may consider the set PM = { x = F( e) e IR n+1 : e B n } as a natural representation of the function F. If n = 2, then e = e(u 1, u 2 ) = (cos u 1 cos u 2, cos u 1 sinu 2, sin u 1 ) for (u 1, u 2 ) R = ( π/2, π/2) (0, 2π) and we obtain a potential surface with a parametric representation (Figure 7.1) PS = { x = f(u 1, u 2 )(cos u 1 cos u 2, cos u 1 sinu 2, sin u 1 ) : (u 1, u 2 ) R} where f(u 1, u 2 ) = F( e(u 1, u 2 )). 77

78 Figure 7.1 A potential surface 78

79 Representations of potential surfaces and corresponding Wulff s crystals Figure 7.2 Potential surfaces and corresponding Wulff s crystals 79

80 Representations of potential surfaces and corresponding Wulff s crystals Figure 7.3 Potential surfaces and corresponding Wulff s crystals 80

81 Wulff s principle Let denote the usual inner product in IR n. Wulff gave a geometric principle of construction for crystals. Theorem 7.1 (Wulff s principle) ([8, p. 88]) For every e B n, let E e denote the hyperplane orthogonal to e and through the point P with position vector p = F( e) e, and H e be the half space which contains the origin and has the boundary E e = H e. Then the crystal C F which has F as its surface energy function is uniquely determined and given by C F = H e = { x : x e F( e)}. e B n e B n Remark 7.2 It is clear that if the surface energy function F is continuous then C F is a closed convex subset of IR n+1. 81

82 Theorems on Wulff s construction Theorem 7.3 ([8, Satz 6.1]) Let F : B n IR + be a continuous function. Then a point X with position vector x is on the boundary C F of Wulff s crystal C F corresponding to F if and only if (left in Figure 7.4) F( e) x e for all e B n and F( e 0 ) = x e 0 for some e 0 B n. Theorem 7.4 ([8, Satz 6.2]) Let F : B n IR + be a continuous function and CF : B n IR + be defined by { } (7.6) CF( e) = inf F( u)( e u) 1 : u B n and e u > 0. Then the boundary C F of Wulff s crystal corresponding to F is given by (7.7) C F = { x = CF( e) e IR n+1 : e B n }. 82

83 Wulff s crystal for n = 2 Example 7.5 If n = 2, then a parametric representation for the boundary C F of Wulff s crystal corresponding to F is (right in Figure 7.4) (7.8) x(u 1, u 2 ) = CF( e(u 1, u 2 )) e(u 1, u 2 ) for (u 1, u 2 ) 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

84 Potential surfaces and Wulff s crystals constructed by Theorems 7.1 and 7.3 Figure 7.5 Wulff s crystals constructed by Theorems 7.3 and

85 The special case F = When F is equal to a norm in IR 3, then the boundary of the corresponding Wulff s crystal is given by the dual norm of. Corollary 7.6 Let be a norm on IR n+1 and, for each w B n, let φ w :IR n+1 IR be defined by φ w (x) = w x = n+1 w k x k ( x IR n+1 ). Then the boundary C of the corresponding Wulff s crystal is given by { (7.9) C = x = 1 } φ e e IRn+1 : e B n, where φ e is the norm of the functional φ e, hence the dual norm of. 85

86 The dual cases F = 1 and F = Figure 7.6 Wulff s crystals corresponding to the norms 1 and 86

87 The case F = w p Figure 7.7 Potential surface of the w p norm and potential surface with corresponding Wulff s crystal 87

88 The case F = Mp Figure 7.8 Potential surface of the M p norm and potential surface with corresponding Wulff s crystal 88

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92 REFERENCES Previous Next First Last Back Zoom To Fit FullScreen Quit REFERENCES [27] A. Wilansky, Summability Through Functional Analysis, North-Holland Mathematics Studies 85, Amsterdam, , 15, 16, 17, 21, 22, 23, 24, 27, 29, 56 [28] G. Wulff, Der Curie Wulffsche Satz über Combinationsformen von Krystallen, Z. Krystallogr. 53 (1901) 76 92