On Statistical and Ideal Convergence of Sequences of Bounded Linear Operators

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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 3, 2015, 357-367

ISSN 1307-5543 – www.ejpam.com

On Statistical and Ideal Convergence of Sequences of Bounded

Linear Operators

Enno Kolk

Institute of Mathematics, University of Tartu, 50090 Tartu, Estonia

Abstract. Let(A_n)be a sequence of bounded linear operators from a separable Banach spaceX into a Banach spaceY. Suppose thatΦis a countable fundamental set ofXand the idealI of subsets ofN_has

property (AP). The sequence(An)is said to beb∗I-convergent if it is pointwiseI-convergent and there

exists an index setK such thatN\_K∈ I _and_(A_k_x)_k∈K _{is bounded for any} _x∈_X_{. We prove that the}

sequence(A_n)isb∗I-convergent if and only if(kAnk)isI-bounded and(Anφ)isI-convergent for any φ∈Φ. Applications of this Banach–Steinhaus type theorem are related to some sequence-to-sequence matrix transformations and to the weakI-convergence in Banach spaces.

2010 Mathematics Subject Classifications: 40A35; 40C05, 40J05, 46B15, 46B45

Key Words and Phrases: Banach space, bounded linear operator, ideal,I-convergence, I-boundedness, matrix method of summability, sequence space, statistical convergence, weakI-convergence, weak statistical convergence

1. Introduction and Preliminaries

LetN₌{_{1, 2, . . .}}_{and let}_X_,_Y _{be two normed spaces over the field}K_{of real numbers}R

or complex numbersC_{. A subset}ΦofX is called fundamental if the linear span ofΦis dense inX. ByB(X,Y)we denote the space of all bounded linear operators fromX intoY. As usual, the dual ofX is defined byX′=B(X,K₎_{. By}_ω₍_X₎_{we denote the set of all}_X_{-valued sequences.}

We write sup_n, limn and

P

n instead of supn∈N, limn→∞andP∞n=1, respectively. By anindex setwe mean any infinite set{k_i} ⊂N_with_k_i_<_k_i₊₁ _{for each}_i∈N_.

Let A_n ∈B(X,Y) (n∈N₎_{. The following theorems of functional analysis are well known}

(see, for example,[11]or[17]).

Theorem 1(Principle of uniform boundedness). Let X be a Banach space. Ifsup_nkAnxk<∞

for every x∈X , then

sup

n

kA_nk<∞. (1)

Email address:[email protected]

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Theorem 2(Banach–Steinhaus). Let X , Y be two Banach spaces and let Φ be a fundamental set of X . The limit lim_nA_nx exists for any x ∈ X if and only if (1)holds and lim_nA_nφ exists for every φ ∈Φ. Moreover, the limit operator A0, A0x = limnAnx is bounded and linear, i.e.,

A₀∈B(X,Y), and kA₀k ≤sup_nkA_nk. If A∈B(X,Y), thenlim_nA_nx =Ax for any x∈X if and only if (1)holds andlim_nA_nφ=Aφ(φ∈Φ).

The first idea of statistical convergence appeared, under the name of almost convergence, in the first edition (Warsaw, 1935) of the monograph[25] of Zygmund. Since 1951 when Fast[7] (see also[23] and[22]) introduced statistical convergence of number sequences in terms of asymptotic density of subsets ofN_{, several applications and generalizations of this}

notion have been investigated (for references see[4] and[6]). For instance, Maddox [20]

and Kolk[13] considered the statistical convergence of sequences taking values in a locally convex space or a normed space, respectively. An another extension of statistical convergence is related to generalized densities.

Let T = (tnk) be a non-negative regular matrix of scalars (i.e., tnk ≥ 0 (n,k ∈N) and

lim_nP_kt_nku_k=lim_ku_kfor any convergent scalar sequence(u_k)). A setK⊂N_{is said to have}

T-densityδ_T(K)if the limit

δ_T(K) =lim

n

X

k∈K

tnk

exists (cf.[9]).

A sequence x = (x_k) ∈ ω(X)is called T -statistically convergentto a point l ∈ X, briefly st_T-limx_k=l, if

δ_T({k:kxk−lk ≥ǫ}) =0

for everyǫ >0 (see[3, Definition 7]and[14, p. 44]).

IfTis the identity matrixI, thenT-statistical convergence is just the ordinary convergence inX and if T is the Cesàro matrixC₁, then T-statistical convergence is statistical convergence as defined by Fast[7].

A further extension of statistical convergence was given in[16]by means of ideals. Recall that a subfamilyI of the family 2N

of all subsets ofN _{is called an}_ideal_{if for each} _K,_L∈ I

we haveKSL∈ I and for eachK∈ I and each L⊂K we haveL∈ I. An idealI is called non-trivialifI 6=;andN∈ I/ . A non-trivial idealI is calledadmissibleifI contains all finite subsets ofN_{. Any non-trivial ideal}I _{defines a}_filter

F(I) ={K⊂N_:N\_K∈ I }_.

For example,

I_T ={K⊂N_:_δ_T₍_K_{) =}₀}

is an admissible ideal and theI_T-convergence coincides with theT-statistical convergence. An admissible ideal I ⊂ 2N

is said to have property (AP) if for every countable family of mutually disjoint sets K1,K2, . . . fromI there exist sets L1,L2, . . . from 2

N

such that the symmetric differencesK_i∆L_i (i∈N₎_{are finite and}_L₌S

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Remark 1([1], Proposition 1). The property (AP)is equivalent to the property (P): for every countable family of sets K1,K2, . . . fromI there exist a set K∈ I such that the differences Ki\K (i∈N₎_{are finite.}

A sequence x = (xk) ∈ω(X) is said to beI-convergent to l ∈X, brieflyI-limkxk = l,

if for eachǫ >0 the set{k∈N_:k_x_k−_lk ≥_ǫ}_{belongs to}I _[_{16, Definition 3.1}_]_{. With the}

I-convergence are closely related the following two notions. A sequencex= (x_k)∈ω(X)is said to be I∗-convergenttol ∈X, brieflyI∗-limxk= l, if there exists an index set K = (ki)

such thatK ∈ F(I)and lim_ix_k_i =l inX [16, Definition 3.2]). A sequencex= (x_k)∈ω(X)

is said to beI-bounded, briefly x_k = O_I(1), if there exists an index set K = (k_i) such that K ∈ F(I)and the sequence(ki)is bounded in X (cf. [10]). In the special caseI =IT we

writeO_st_T(1)instead ofO_I(1).

We remark that theI∗-convergence of number sequences was introduced already by Freed-man[8]asI-near convergence.

It is easy to see that I∗-convergence implies I-convergence and every I∗-convergent sequence isI-bounded.

The following characterization ofI-convergence is important for us.

Proposition 1([16, Theorem 3.2]). If the idealI has property(AP), thenI-limx_k = l in a Banach space X if and only ifI∗-limxk=l.

BycI(X)we denote the set of allI-convergentX-valued sequences. Letℓ∞(X),c(X)and c0(X) be the sets of all bounded, convergent and convergent to zero X-valued sequences, respectively. For 1 ≤ p < ∞ let ℓ_p(X) be the set of sequences (x_k) ∈ ω(X) such that P

kkxkkp<∞.

Using Proposition 1 and Theorem 2, we proved in [15] the following Banach–Steinhaus type theorem forI-convergence.

Theorem 3([15, Theorem 3]). Let X and Y be two Banach spaces, where X has a countable fundamental setΦ. If the idealI has property(AP), then the sequence(A_n) is bI-convergent

(i.e.,(Anx)∈cI(Y)∩ℓ∞(Y)for any x∈X)if and only if (1)holds and(Anφ)isI-convergent

for every φ ∈ Φ. Thereby, the limit operator A, Ax = I-limA_nx, is bounded and linear, and kAk ≤sup_nkA_nk.

In this paper we introduce the notion of b∗I-convergence of sequences of bounded lin-ear operators (A_n) and give an analogue of Theorem 3 by finding necessary and sufficient conditions forb∗I-convergence of such sequences(An). As applications of this result we

char-acterize infinite summability matricesA= (Ank)of typeA:λ(X) b∗I

−→c(Y)withAnk∈B(X,Y) (n,k∈N)andλ∈ {c, c0, ℓ1}, also consider the weak b∗I-convergence in Banach spaces.

2. Main Theorems

In the following letX,Y be two Banach spaces,A_n∈B(X,Y) (n∈N₎_{and let}I ⊂₂N_{be a}

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Recall that a sequence(x_n)∈ω(X)is said to beweaklyI-convergent(weakly T -statistically convergent) to a pointl ∈X ifI-limx′(x_n) = x′(l)(st_T-limx′(x_n) = x′(l)) for any x′ ∈X′

[2, 21]. We know that every weakly convergent sequence in a Banach spaceX is bounded. But a weaklyI-convergent sequence is not necessaryI-bounded (cf. [5, Theorem 1]). Example 2 from[5]shows that the Banach sequence spaceℓ₂contains a weakly statistically null sequence

(zk)with no bounded subsequences. Thus some results of Bhardwaj and Bala[2, Theorem 3.1

and Lemma 3.2] are incorrect. At it, defining F_nx′ = x′(z_n) (x′ ∈ ℓ′₂, n ∈ N₎_{, we get the}

sequence(F_n)of bounded linear functionals F_n :ℓ′₂ →R_{. Since}k_F_nk₌k_z_nk_{by the classical}

Hahn–Banach theorem, the sequence of functionals(Fn)converges statistically to zero for any

x′∈ℓ′₂, but the sequence of norms(kF_nk)contains no bounded subsequences. This example justifies the following definition.

Definition 1. A sequence(A_n)of operators A_n∈B(X,Y) (n∈N₎_{is said to be b}∗I_{-convergent (to}

A∈B(X,Y)) ifI-lim_nA_nx exists (I-limA_nx=Ax) for any x ∈X and there is a set K∈ F(I)

such that(Akx)k∈K is bounded for every x ∈X . In the special case I =IT we get the notion

of b∗T -statistical convergence. The b∗I-limit and the b∗T -statistical limit of (A_n)are denoted, respectively, by b∗I-lim_nA_nand b∗st_T-lim_nA_n.

In view of Theorem 1 we can say that a sequence (A_n) is b∗I-convergent if and only if I-limA_nx exists for any x∈X and

sup

k∈K

kA_kk)<∞for someK∈ F(I). (2)

Theorem 3 shows that bI-convergence implies b∗I-convergence by the suppositions that X is separable andI satisfies the condition (AP).

To prove our main theorem we need the following lemma.

Lemma 1. Suppose that the idealI has property(AP)and let z_{k j}∈X (k,j∈N).

IfI-lim_kz_{k j}=z_jfor any j∈N_{, then there exists an index set N} _{= (}_n_i₎_{such that N} ∈ F₍I₎_and

lim_izni,j=zjfor any j∈N.

Proof. Assume thatI-lim_kz_{k j} =z_j (j∈N₎_{. Since}I _{has property (AP), by Proposition 1}

there exist index setsKj={ki(j)}(j∈N)such that

lim

i zki(j),j=zj (j∈N) (3)

andK′_j = N\_K_j ∈ I _{for any} _j ∈N_{. Because of Remark 1 we can find the set} _N′ ∈ I _such

that the differencesK′_j\N′ (j ∈N)are finite. Now, forN =N\_N′ _{we have that} _N ∈ F(I)

and the differencesN\K_jare finite. Consequently, denotingN= (n_i), from (3) it follows that lim_iz_n

i,j=zjfor any j∈N.

Theorem 4. Let X and Y be two Banach spaces, where X has a countable fundamental setΦ. If the idealI has property(AP). A sequence(A_n)of operators A_n∈B(X,Y)is b∗I-convergent if and only if(kAnk)isI-bounded, i.e.,(2)holds, and(Anφ)isI-convergent for everyφ∈Φ. Thereby,

the limit operator A₀, A₀x = I-limA_nx, is bounded and linear, and kA₀k ≤ sup_k_∈_KkA_kk. If A∈B(X,Y), then b∗I-lim_nA_n = A if and only if(kA_nk) isI-bounded andI-lim_nA_nφ=Aφ

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Proof. If (A_n)is b∗I-convergent (b∗I- lim_nA_n = A), then (2) is satisfied andI-limA_nφ

exists (I-limA_nφ=Aφ) for everyφ∈Φ.

Conversely, assume that (2) holds andI-limAnφj exists (orI-limAnφj=Aφj) for every

j ∈N_{, where}_{Φ =}{_φ_j}_{. Applying Lemma 1 to} _z_{n j} ₌ _A_n_φ_j _(and _z_j ₌_A_φ_j_{), we fix an index}

set N = (n_i)∈ F(I)such that lim_iA_n

iφj exists (limiAniφj =Aφj) for any j∈N. Since the set M = N∩K also belongs to F(I), denoting M = (mi), we have that limiAmiφj exists (lim_iA_m_iφ_j=Aφ_j) for any j∈N_{and sup}_ik_A_m

ik<∞. So, by Theorem 2, the limit A₀x =lim_iA_m

ix exists (limiAmix =Ax) for anyx ∈X, A0 ∈B(X,Y)andkA0k ≤supikAmik. The proof is completed if we remark that limiAmix =I- limAnx by Proposition 1.

It is known that the ideal I_T = {K ⊂ N_:δ_T(K) =0}defined by a non-negative regular matrix T has the property (AP) (see [9, Proposition 3.2]). Since I_T-convergence coincides with T-statistical convergence, from Theorem 4 we immediately get the following Banach– Steinhaus type theorem for b∗T-statistical convergence.

Theorem 5. Suppose that T is a non-negative regular matrix and X has a countable fundamental setΦ. A sequence(An)of operators An∈B(X,Y)is b∗T -statistically convergent if and only if (2)

holds and st_T-limA_nφexists for anyφ∈Φ. In this case the limit operator A₀, A₀x=st_T-limA_nx

(x ∈X), belongs to B(X,Y)andkA₀k ≤sup_k_∈_KkA_kk. If A∈B(X,Y), then b∗st_T-lim_nA_n=A if and only if(kAnk)isI-bounded and stT-limnAnφ=Aφ(φ∈Φ).

3. Some Applications

Let λ(X) be a subspace of ω(X), µ(Y) a subspaces of ω(Y) and A = (Ank) an infinite

matrix of operatorsA_nk∈B(X,Y) (n,k∈N₎_{. We say that}_A_maps_λ₍_X₎_into_µ₍_Y₎_{, and write}

A:λ(X)→µ(Y), if for allx= (x_k)∈λ(X)the seriesAnx=

P

kAnkxk (n∈N)converge and

the sequenceAx= (Anx)belongs toµ(Y).

It is well known thatc(X),c₀(X)andℓ_∞(X)are Banach spaces with the norm kxk_∞=sup_kkx_kk, andℓ_p(X)is Banach space with the normkxk_p= P_kkx_kkp1/pif 1≤p<∞.

For x ∈ X and n ∈ N _let _e₍_x_{) = (}_x_,_x_{, . . .}₎ _{be constant sequence and} _ek₍_x_{) = (}_ek

j(x))

the sequence with ek_j(x) = x if j = k and ek_j(x) = 0 otherwise. It is not difficult to see that if Φ is a (countable) fundamental set in X, then E₀(Φ) = {ek(φ) : k ∈ N_, _φ ∈ _Φ} _is

a (countable) fundamental set in Banach spaces c0(X) and ℓp(X), andE0(Φ) S

E1(Φ) with E₁(Φ) ={e(φ):φ∈Φ}is a (countable) fundamental set in Banach spacec(X).

Using Theorem 2, Zeller[24](see also[19]) and Kangro[12]characterized the matrices A:c(X)→c(Y),A:c0(X)→c(Y)andA:ℓ1(X)→c(Y)as follows.

Theorem 6. LetA= (A_nk)be an infinite matrix with A_nk∈B(X,Y). Then: (i) A:c(X)→c(Y)if and only if

G_n=sup

r

sup kxkk≤1

r

X

k=1 A_nkx_k

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sup

n

G_n<∞, (5)

∃lim

n Ankx (k∈

N_, _x∈_X₎_, ₍₆₎

∃lim

m m

X

k=1

Ankx (n∈N, x ∈X), (7)

∃lim

n

X

k

A_nkx (x ∈X); (8)

(ii) A:c₀(X)→c(Y)if and only if (4)–(6)hold; (iii) A:ℓ₁(X)→c(Y)if and only if (6)is satisfied and

H_n=sup

k

A_nk

<∞ (n∈N), (9) sup

n

H_n<∞,

Remark 2. It is not difficult to see, using Theorem 2, that in Theorem 6 it suffices to require the fulfillment of conditions(6)–(8)for all elementsφfrom a fundamental setΦof X .

The notion of b∗I-convergence of sequences of bounded linear operators leads us to the definition of new type summability maps.

Definition 2. Letλ(X)andµ(Y)be two linear subspaces ofω(X)andω(Y), respectively, and letI ⊂2N be a non-trivial admissible ideal. We say that a matrixA mapsλ(X)in the sense of b∗I-convergence intoµ(Y), and writeA:λ(X)b

∗_I

−→µ(Y), ifI-limA_nx exists for anyx∈λ(X)

and there is an index set N= (n_i)fromF(I)such that the submatrixA(N)= (ani,k)mapsλ(X)

intoℓ_∞(Y). In the case ofI =I_T we get the matrices of typeA:λ(X)b ∗_st

T −→µ(Y).

Based on Theorems 4 and 5, we describe the matricesA:λ(X)b ∗_I

−→c(Y)and A:λ(X)b

∗_st

T

−→c(Y), whereλ∈ {c,c₀,ℓ_p}.

Proposition 2. LetA= (A_nk)be an infinite matrix with A_nk∈B(X,Y). Suppose that X has a countable fundamental setΦand the idealI has property(AP). Then:

(i) A:c(X)b ∗_I

−→c(Y)if and only if (4)and(7)hold,

Gn=OI(1), (10)

∃I-lim

n Ankφ (k∈

N_, _φ∈_Φ)_, ₍₁₁₎

∃I-lim

n

X

k

A_nkφ (φ∈Φ); (12)

(ii) A:c0(X)

b∗I

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(iii) A:ℓ₁(X) b ∗_I

−→c(Y)if and only if (9)is satisfied and(H_n)isI-bounded. Proof. The equalityA(_nr)x=Pr

k=1Ankxk defines a linear operatorAn(r) onc(X)and c0(X) for anyn,r ∈N_{. Since}

kA(r)

n k=

r X k=1 A_nkx_k

,

by Theorem 2 we get that the seriesA_nx(n∈N₎_{converge for all}_x∈_c₍_X₎_and_A_n∈_B₍_c₍_X₎_,_Y₎

if and only if (4), (7) are satisfied. Similarly,An ∈B(c0(X),Y)if and only if (4) holds. Now, applying Theorem 4 to the operatorsAn, we have thatA:c(X)

b∗I

−→c(Y)(orA:c₀(X) b ∗_I

−→c(Y)) if and only if (10) holds and(Any)isI-convergent for anyy∈ E1(Φ)(respectively,y∈ E0(Φ)). But this reduces to (11) and (12) becauseA_ne_k(φ) =A_nkφandA_ne(φ) =P

kAnkφ.

SinceAn ∈B(ℓ1(X),Y)if and only if (9) holds, the statement (iii) also follows by Theo-rem 4.

The matrix mapA:ℓ_p(X)b ∗_I

−→c(Y)we consider in the special casesY =K_{and 1}_<_p_<∞_.

Then B(X,Y) = X′ and so, A_nk ∈ X′ (n,k ∈ N₎_{. In this case} _A_n ∈ ₍_ℓ_p₍_X₎₎′ _{if and only if}

(A_nk)_k_∈N∈ℓ_q(X′), i.e.,

P

kkAnkkq<∞, where 1/p+1/q=1. Therefore, denotingc=c(K)

and using the same arguments as in the proof of Proposition 2, we get the following result. Proposition 3. Let A = (A_nk) be an infinite matrix with A_nk ∈ X′. Suppose that X has a countable fundamental set Φ, the idealI has property(AP)and1< p< ∞, 1/p+1/q= 1. ThenA:ℓ_p(X)b

∗_I

−→c if and only if (11)holds and X

k

kA_nkkq=OI(1).

If X =Y =K_{, then the matrix map}_A_{reduces to the transformation}_A_:_λ→_µ_{defined by}

an infinite scalar matrixA= (ank). Using the fact that forY =Kwe have (see[12, p. 114])

sup kxkk≤1

r X k=1 Ankxk

= r X k=1 kAnkk,

from Propositions 2 and 3 we obtain the following corollary.

Corollary 1. Let A= (a_nk)be an infinite matrix of scalars,1< p<∞and1/p+1/q=1. If the idealI has property(AP), then:

(i) A:c b

∗_I

−→c if and only if

X

k

|ank|=OI(1), (13)

∃I-lim

n ank (k∈

N₎_, ₍₁₄₎

∃I-lim

n

X

k

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(ii) A:c₀ b

∗_I

−→c if and only if (13)and(14)hold;

(iii) A : ℓ₁ b

∗_I

−→c if and only if (14) is satisfied, h_n = sup_k|a_nk| < ∞ (n ∈ N₎ _and ₍_h_n₎ _is

I-bounded;

(iv) A:ℓ_p b

∗_I

−→c if and only if (14)is satisfied and X

k

|a_nk|q=OI(1).

LettingI =IT in Propositions 2, 3 and Corollary 1, we get the characterizations of

ana-logical matrix maps in the sense ofb∗T-statistical convergence. We also remark that the matrix maps in the sense of bI- and bst_T-convergence were studied in[15].

At the beginning of Section 2 we remarked that a weakly I-convergent sequence is not necessaryI-bounded. This fact leads us to a new variant of weakI-convergence.

Definition 3. A sequencex= (x_n)∈ω(X)is said to be weakly b∗I-convergent to l ∈X , briefly wb∗I-limnxn = l, ifx is weaklyI-convergent to l and there is a set K ∈ F(I)such that the

sequence (x′(x_k))_k_∈_K is bounded for every x′ ∈ X′. For I = I_T we get the notion of weak b∗T -statistical convergence, in this case we write wb∗st_T-lim_nx_n=l.

Using bounded linear functionals F_z :X′→R_, _F_z_x′₌ _x′₍_z_{) (}_x′∈_X′_,_z∈_X₎_{, we can say}

thatwb∗I- lim_nx_n=l(wb∗st_T- lim_nx_n=l) if and only if the sequence(F_x n)isb

∗_I_-convergent (b∗T-statistically convergent) to Fl. Thus, sincekFzk=kzk, by Theorems 4 and 5 we get the

following characterizations of these new types of weak convergence.

Proposition 4. Letx= (x_n)∈ω(X)and l ∈X . Assume that X′has a countable fundamental setΦ′.

(i) IfI is an ideal with the property(AP), then wb∗I-lim_nx_n=l if and only if

kx_nk=OI(1), (15)

I-lim

n φ

′₍_x

n) =φ′(l) (φ′∈Φ′). (16)

(ii) If T is a regular matrix, then wb∗stT-limnxn=l if and only if (15)and(16)are satisfied

with st_T instead ofI.

Finally we apply Proposition 4 to Banach sequence spacesc0(X)andℓp(X)with

1<p<∞. It is known that the dual spacesc0(X)′andℓp(X)′are isometrically isomorphic,

respectively, to ℓ₁(X′)and ℓ_q(X′), where 1/p+1/q = 1 (see, for example,[18]). If Φ′ is a fundamental set of X′, thenE0(Φ′) is the fundamental set ofℓ1(X′) andℓq(X′). Thus from

Proposition 4 we get the following two corollaries.

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(i) IfI is an ideal with the property(AP), then wb∗I-lim_nxn=x0 if and only if

kxnk∞=OI(1), (17)

I-lim

i φ

′₍_x

ni) =φ′(xi) (φ′∈Φ′,n∈N). (18)

(ii) If T is a non-negative regular matrix, then wb∗st_T-lim_nxn=x0 if and only(17)and(18) hold with st_T instead ofI.

Corollary 3. Letxn = (xni) (n ∈N) and x0 = (xi) be the elements of ℓp(X) (1< p < ∞).

Assume that X′has a countable fundamental setΦ′.

(i) IfI is an ideal with the property(AP), then wb∗I-lim_nxn=x0 if and only if (18)is true and

kx_nk_p=O_I(1). (19)

(ii) If T is a non-negative regular matrix, then wb∗stT-limnxn=x0 if and only(18)and(19) hold with st_T instead ofI.

Proposition 4(ii) and Corollary 3(ii), for T = C1, may be considered as some corrected versions, respectively, of Theorem 3.1 and Lemma 3.2 from[2].

ACKNOWLEDGEMENTS The author is thankful for the support by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.

References

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