Statistical convergence and some questions of operator theory.

1. Introduction and Background

In this paper, we use the concept of statistical convergence for solving of some problems of operator theory.

Recall that the concept of statistical convergence was firstly introduced by Fast in [4], see also Steinhaus [14]. In what follows statistical convergence studied in many further papers (see, for instance, Fridy [5, 6], Kolk [8], Pehlivan and Karaev [11], Connor et al. [2]). Following [9], note that if S is a subset of the positive integers N, then [sub S]n denotes the set {s [member of] S : s [less than or equal to] n} and [absolute value of [sub S]n] denotes the number of elements in Sn. The natural density of S by Niven and Zuckerman [9] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.1. A sequence [([x.sub.k]).sub.k[greater than or equal to]1] of real (or complex) numbers is said to be statistically convergent to some number L, if for each [epsilon] > 0 the set

[S.sub.[epsilon]] := {k [member of] N: [absolute value of [x.sub.k] - L] [greater than or equal to] [epsilon]} has natural density zero; in this case we abbreviate [st-lim.sub.k] [x.sub.k] = L.

We recall that (see Fridy [5]) for two sequences x = [([x.sub.k]).sub.k[greater than or equal to]1] and [([y.sub.k]).sub.k[greater than or equal to]1] the notion "[x.sub.k] = [y.sub.k] for almost all k" means that [delta]({k : [x.sub.k] [not equal to] [y.sub.k})] = 0. The following is the classical result of Fridy [5].

Lemma 1.2. The following statements are equivalent:

(i) [([x.sub.k]).sub.k[greater than or equal to]1] is a statistically convergent sequence;

(ii) [([x.sub.k]).sub.k[greater than or equal to]1] is a sequence for which there is a convergent sequence [([y.sub.k]).sub.k[greater than or equal to]1] such that [x.sub.k] = [y.sub.k] for almost all k.

An immediate and useful corollary of this lemma is the following.

Corollary 1.3. If [([x.sub.k]).sub.k[greater than or equal to]1] is a sequence such that [st-lim.sub.k] [x.sub.k] = L, then [([x.sub.k]).sub.k[greater than or equal to]1] has a subsequence [([y.sub.k]).sub.k[greater than or equal to]1] such that [lim.sub.k] [y.sub.k] = L (in the usual sense).

The notion of statistically convergence was extended to the sequences of Banach spaces by Connor, Ganichev and Kadets in their paper [2] as follows.

Definition 1.4 ([2]). Let X be a Banach space, [([x.sub.k]).sub.k[greater than or equal to]1] be a X-valued sequence, and x [member of] X be an element.

i) The sequence [([x.sub.k]).sub.k[greater than or equal to]1] is norm statistically convergent to x provided that

[delta]({k : [[parallel][x.sub.k] - x[parallel].sub.X] > [epsilon]}) = 0 for all [epsilon] > 0.

(ii) The sequence [([x.sub.k]).sub.k[greater than or equal to]1] is weakly statistically convergent to x provided that, for any continuous linear functional f on X, the sequence [(f ([x.sub.k] - x)).sub.k[greater than or equal to]1] is statistically convergent to 0.

Note that similar to the sequences of numbers, if a Banach space valued sequence x = [([x.sub.k]).sub.k[greater than or equal to]1] is norm statistically convergent, then there exists a usual convergent sequence y = [([y.sub.k]).sub.k[greater than or equal to]1] such that [x.sub.k] = [y.sub.k] for almost all k, i.e., [delta] ({k : [x.sub.k] [not equal to] [y.sub.k]}) = 0 (see [2]). As a consequence, many of the results for real statistically convergent sequences carry over to norm statistically convergent sequences (see Kolk [8]). It is also natural to define a series [[summation].sub.k] [x.sub.k] to be norm statistically convergent to x by requiring the sequence of partial sums ([[summation].sub.n.sup.k=1] [x.sub.k])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be norm statistically convergent to x.

2. Statistical convergence and compactness of operators

The following definition is well known (see, for example, in Pehlivan and Karaev [11]).

Definition 2.1. The sequence [([T.sub.n]).sub.n[greater than or equal to]1] [member of] B (H) (Banach algebra of bounded linear operators on the Hilbert space H) is called weakly statistically convergent to T [member of] B (H) if ([T.sub.n]x, y) statistically converges to (Tx, y) for any x, y [member of] H.

Here we will interested with the following question: if [([T.sub.n]).sub.n[greater than or equal to]1] [member of] B (H) is a sequence of compact operators weakly statistically converging to the operator T, then under which additional conditions T is also compact?

Note that for the detail of this question for the usual weakly convergent sequences of compact operators, the reader can be consult in the paper by Karaev [7].

In this section, we will prove a positive result under some assumption on the so-called Berezin symbols of compact operators [T.sub.n]. So, let us first introduce some necessary notations and preliminaries.

Recall that a Reproducing Kernel Hilbert Space (RKHS) H = H([OMEGA]) is the Hilbert space of complex-valued functions on some set Q with nonempty boundary [partial derivative][OMEGA] such that the evaluation functionals f [right arrow] f ([lambda]), [lambda] [member of] [OMEGA], are continuous H. Then by the classical Riesz theorem about representation of linear continuous functionals on the Hilbert space, for each [lambda] [member of] [OMEGA] there exists a unique function [k.sub.H,[lambda]] [member of] H such that

<f, [k.sub.H,[lambda]>] = f ([lambda])

for all f [member of] H. The functions [k.sub.H,[lambda]] (z), [lambda] [member of] [OMEGA], are called the reproducing kernels of the space H. It is well-known that (see Aronzajn [1] and Saitoh [12, 13]) the reproducing kernel [k.sub.H,[lambda]] of H is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any orthonormal basis [(en(z)).sub.n[greater] than or equal to]1 of the space (separable) H([OMEGA]).

For example, the classical Hardy, Bergman and Fock spaces are RKHSs. Following Nordgren and Rosenthal [10], we say that RKHS H(Q) is standard if the underlying set [OMEGA] is a subset of a topological space and the boundary [partial derivative][OMEGA] is non empty and has the property that the normalized reproducing kernel [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges weakly to 0 whenever [([[lambda].sub.n]).sub.k[greater than or equal to]1] [member of] [OMEGA] converges to any point in [partial derivative][OMEGA].

Note that the most of RKHS, including Hardy, Bergman and Fock Hilbert spaces, are standard in this sense. (Also note that every finite dimensional Hilbert space is non-standard, because in the finite dimensional space the weak and strong conver gence coincide.)

For any bounded operator T : H [right arrow] H its Berezin symbol (see, Nordgren and Rosenthal [10] and Zhu [15]) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is clear that [absolute value of [??]([lambda])] [less than or equal to] [parallel]T[parallel] for all [lambda] [member of] [OMEGA] and hence [??] is a bounded function on [OMEGA]. It is also easy to see that for every compact operator T on the standard RKHS H its Berezin symbol [??] vanish on the boundary.

The following result of Nordgren and Rosenthal [10, Corollary 2.8] characterizes compact operator on the standard RKHS H, in terms of boundary behavior of Berezin symbols its unitary orbits [U.sup.-1]TU, U [member of] B (H).

Lemma 2.2. Let T [member of] B (H) be an operator on the standard RKHS H = H([OMEGA]). Then T is compact if and only if for every unitary operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] tends to 0 whenever [lambda] tends to the boundary points of [partial derivative][OMEGA].

The main result of this section is the following, which generalizes a result of the paper [7, Theorem 4.1].

Theorem 2.3. Let H = H([OMEGA]) be a standard RKHS on some set Q and [([T.sub.n]).sub.n[greater than or equal to]1] a sequence of compact operators on H weakly statistically converging to an operator T [member of] B (H). If the double statistical limit

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exists for every unitary operator U [member of] B (H), then T is compact and l = 0 holds.

Proof. By condition, [T.sub.n] [right arrow] T (n [right arrow] [infinity]) weakly statistically, which means that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f, g [member of] H. In particular, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is any sequence tending to a point in [partial derivative][OMEGA], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any m [greater than or equal to] 1. On the other hand, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every unitary operator U [member of] B (H) and m [greater than or equal to] 1. Since [T.sub.n], n [greater than or equal to] 1, are compact operators and H is standard, by Lemma 2.2 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any unitary operator U [member of] B (H).

Now by considering condition (1) of the theorem, and also the equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any unitary operator U [member of] B (H). Then, by Corollary 1.3, there exists a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [member of] [OMEGA] tending to a point in [partial derivative][OMEGA] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any unitary operator U [member of] B (H). So, by Lemma 2.2 we assert that T is compact, which proves the theorem.

3. Statistical convergence and Schatten-Neumann class operators

Recall that if T is a compact operator on a separable Hilbert space H, then there exists orthonormal sets [([u.sub.n]).sub.n[greater than or equal to]0] and [([v.sub.n]).sub.n[greater than or equal to]0] in H such that

Tx = [[infinity].summation over (n=0)] [[lambda].sub.n] <x, [u.sub.n]> [v.sub.n], x [member of] H,

where [[lambda].sub.n] is the nth singular value (s-number) of T. Given p [member of] (0, [infinity]), we define the Schatten-Neumann p-class of H, denoted by [[??].sub.p] (H) or simply [[??].sub.p], to be the space of compact operators T on H with its singular value sequence [([[lambda].sub.n]).sub.k[greater than or equal to]1] belonging to [l.sup.p] (the p-summable sequences space). We will only consider the case 1 [less than or equal to] p + [infinity] since in this case [[??].sub.p] is a Banach space with the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The class [[??].sub.1] is also called the trace class of H (or nuclear operator class) and [[??].sub.2] is called the Hilbert-Schmidt class. It is easy to show that if T is a compact operator on H and p [greater than or equal to] 1, then T [member of] [[??].sub.p] if and only if [[absolute value of T].sup.p] := [(T*T).sup.p/2] [member of] [[??].sub.1] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our following result improves and generalizes a similar result of the paper [3] by using the concept of statistical convergence and by considering any RKHS instead of weighted Bergman space [L.sup.2.sub.a] (d[A.sub.[alpha]]); see [3, Lemma 5.2].

Lemma 3.1. Let p [member of] [1, + [infinity]), T [member of] B (H) and [T.sub.n] [member of] [[??].sub.p] for all n [greater than or equal to] 1; where H = H([OMEGA]) is a RKHS on some suitable set [OMEGA]. If Tn weakly statistically converges to T in B (H) and [parallel][T.sub.n] [[parallel].sub.p] [less than or equal to] C < + [infinity] and for some constant C > 0; then T [member of] [[??].sub.p] and [[parallel]T[parallel].sub.p] [less than or equal to] C.

Proof. The proof essentially uses the similar arguments from [3], and we present it here only for the sake of completeness. So, for any n [member of] N, define

[[xi].sub.n] (K) = tr ([T.sub.n]K).

Then we have [[xi].sub.n] [member of] [[??].sup.*.sub.q], where 1-/p + 1/q = 1 And [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By Banach-Alaoglu's theorem, there exists a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in w*-topology and [xi] [member of] [[??].sup.*.sub.q]. Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all K [member of] [[??].sub.q] and [absolute value of [xi](K)] [less than or equal to] M [[parallel]K[parallel].sub.q] for some M > 0. On the other hand, since [T.sub.n] weakly statistically converges to T, we deduce that st-tr ([T.sub.n]K) [right arrow] tr (TK) for all operators K of finite rank. Thus, the lemma follows since

[[parallel]T[parallel].sub.p]= sup {|tr (TK)|: rank (K) < [infinity] and [[parallel]T[parallel].sub.q] [less than or equal to] 1} < [infinity].

Theorem 3.2. Let T G B (H) be an operator on a RKHS H = H ([ohm]), and let T = V [absolute value of T] be its polar decomposition. If T [member of] [[??].sub.p], then V [member of] [[??].sub.p], if 1 [less than or equal to] p < + [infinity].

Proof. Put [T.sub.n] := T ([absolute value of T] + [[alpha].sub.n])-1, where [[alpha].sub.n] > 0, n [greater than or equal to] 1, and [st-limn.sub.[right arrow][infinity]] [[alpha].sub.n] = 0. We will first prove that [st-lim.sub.n[rightarrow][infinity]] [parallel] [T.sub.n] f - Vf [[parallel].sub.H] = 0 for every f [member of] H, that is [T.sub.n] tends to T strongly statistically as n [right arrow][infinity]. For this purpose, let ([E.sub.[lambda]]) be the spectral family for [absolute value of T]. Then, by considering that [st-lim.sub.n[rightarrow][infinity]] [[alpha].sub.n] = 0, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In fact, notice that [absolute value of T] = [[integral].sup.[infinity].sub.0] [lambda]dE[lambda] is the spectral decomposition of [absolute value of T]. We set [A.sub.n] := [absolute value of T] ([absolute value of T] + [[alpha].sub.n] )-1. Then it is clear that

[A.sub.n][E.sub.o]f =([absolute value of T] + [([[alpha].sub.n]).sup.-1] [absolute value of T] [E.sub.o]f = 0

for any f [member of] H, and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [st-lim.sub.n[right arrow][infinity]] [[alpha].sub.n] = 0, by Corollary 1.3 there exists a subsequence [([[beta].sub.n]).sub.n[greater than or equal to]1] of [([[alpha].sub.n]).sub.n[greater than or equal to]1]

such that [lim.sub.n[right arrow][infinity][[beta].sub.n]] = 0. So, from Lebesgue's dominated convergence theorem, Lemma 1.2 and Corollary 1.3, it follows that [A.sub.n] strongly statistically converges to I - [E.sub.o] as n [right arrow] [infinity]. Thus, we obtain that [T.sub.n] [right arrow] V (I - [E.sub.o]) strongly statistically as n [right arrow] [infinity]. Since [E.sub.o] is the projection onto the eigenspace {f G H : Tf = 0}, we have V [E.sub.o] = 0. Consequently, [T.sub.n] [right arrow] V strongly statistically as n [right arrow] [infinity].

Now suppose T [member of] [[??].sub.p]. Then [T.sub.n] [member of] [[??].sub.p], [parallel] [T.sub.n] [[parallel].sub.=] [less than or equal to] C < + [infinity] for some C > 0 and [T.sub.n] [right arrow] V strongly statistically as n [right arrow] [infinity]. By applying Lemma 3.1, we deduce V [member of] [[??].sub.p]. The theorem is proved.

Received April 2, 2015

MEHMET GURDAL, ULAS YAMANCI

Department of Mathematics, Suleyman Demirel University 32260, Isparta, Turkey

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