Characterization of the compact operators on the class (bv, b v k θ)

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Characterization of the compact operators

on the class (bv,

)

Cite as: AIP Conference Proceedings 2183, 050010 (2019); https://doi.org/10.1063/1.5136148

Published Online: 06 December 2019 Fadime Gökçe, and M. Ali Sarıgöl

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Characterization of the compact operators on the class

bv

,bv

θ

_k

Fadime G¨okc¸e

1,a)

and M. Ali Sarıg¨ol

1,b)

1)_{Department of Mathematics, University of Pamukkale, Denizli, Turkey}

a)_{Corresponding author: fgokce@pau.edu.tr} b)_{msarigol@pau.edu.tr}

Abstract. The space bv, the set of all bounded variation sequences, has an important role in the summability theory. In recent study, this spaces has been extended to the space bvθ_k and some matrix class on this space has been characterized [2]. In the present paper, for 1≤ k < ∞, computing Hausdorff measure of non-compactness, we characterize compact operators in the classbv,bvθ_k, where θ is a sequence of positive numbers.

Keywords: Sequence spaces, matrix transformations, bvθkspaces PACS: 40C05, 40D25, 40F05, 46A45

INTRODUCTION

Letω be the set of all complex sequences, lk(1 ≤ k < ∞) and c be the set of all k-absolutely convergent series and convergent sequences, respectively. We write bv=

x= (xk) ∈ w : ∞ ∑ n=0|xn| < ∞

for the set of all sequences of bounded variation. In [2], extending the space bv, the space bvθ_k has been deﬁned by

bvθ_k = x= (xk) ∈ w : ∞

∑

n=0θ k−1 n |xn|k< ∞, x−1= 0 ,

which is a BK space for 1≤ k < ∞, where (θn) is a sequence of nonnegative terms and xn= xn− xn−1for all n. Also, it is reduced to bvkin the special caseθn= 1 for all n, studied by Malkowsky, V.Rakoˇcevi´c and ˇZivkovi´c [1], and bvθ₁ = bv for k = 1.

Let X and Y be subspaces of w and A= (anv) be an arbitrary inﬁnite matrix of complex numbers. By A(x) = (An(x)), we denote the A-transform of the sequence x = (xv), i.e.,

An(x) =

∞

∑

v=0

anvxv,

provided that the series are convergent for v,n ≥ 0. Then, A deﬁnes a matrix transformation from X into Y, denoted by

A∈ (X,Y), if the sequence Ax = (An(x)) ∈ Y for all sequence x ∈ X. Lemma 1 Let 1≤ k < ∞. Then, A ∈ (,k) if and only if

A_(,_k₎= sup v ∞

∑

n=0 |anv|k 1/k < ∞ [1].

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If S and H are subsets of a metric space(X,d) and ε > 0, then S is called an ε-net of H, if, for every h ∈ H, there exists an s∈ S such that d (h,s) < ε; if S is finite, then the ε-net S of H is called a finite ε-net of H. By MX, we denote the collection of all bounded subsets of X. If Q ∈ MX, then the Hausdorff measure of noncompactness of Q is defined by

χ(Q) = {ε > 0 : Q has a ﬁnite ε-net in X}.

The functionχ : MX→ [0,∞) is called the Hausdorff measure of noncompactness [4].

If X and Y are normed spaces,B (X,Y) states the set of all bounded linear operators from X to Y and is also a normed space the norm

L = sup

x∈SX

L(x),

where SXis a unit sphere in X, i.e., SX= {x ∈ X : x = 1}. Further, a lineer operator L : X → Y is said to be compact if its domain is all of X and the sequence(L(xn)) has convergent subsequence in Y for every bounded sequence

x= (xn) ∈ X. We denote the set of such operators by C (X,Y).

The following results are important tool to compute Hausdorff measure of noncompactness.

Lemma 2 Let X and Y be Banach spaces, L∈ B (X,Y). Then, the Hausdorff measure of noncompactness of L, denoted byL_χ, is deﬁned

L_χ= χ (L(SX)),

and

L∈ C (X,Y) iff L_χ= 0 [3].

Lemma 3 Let Q be a bounded subset of the normed space X where X= _kfor 1≤ k < ∞. If Pn: X→ X is the

operator deﬁned by Pr(x) = (x0,x1,...,xr,0,...) for all x ∈ X, then

χ(Q) = lim_r→∞sup

x∈Q(I − Pr)(x),

where I is the identity operator on X [4].

Lemma 4 Let X be normed sequence space,χT andχ denote the Hausdorff measures of noncompactness on

M_XTandMX, the collections of all bounded sets in XTand X, respectively. Then,χT(Q) = χ (T(Q) for all Q ∈ M_XT,

where T is an inﬁnite triangle matrix [3].

Compact operators in the class

bv

,bv

θ_k

The classbv,bvθ_kof inﬁnite matrices has more recently been characterized by Hazar and Sarıg¨ol [2] as follows. In the present paper, by computing Hausdorff measure of noncompactness, we characterize compact operators in the same class.

Theorem 1 Let 1≤ k < ∞ and θ = (θn) be a sequence of positive numbers. Further let A = (anv) be an inﬁnite

matrix of complex numbers for all n,v ≥ 0. Then, A ∈bv,bvθ_kif and only

lim n→∞ ∞

∑

j=ν an jexists (1) sup n,v ∞

∑

j=v an j  < ∞ (2) sup ν ∞

∑

n=0 θ 1−1/k n ∞

∑

j=ν an j− an−1, j k < ∞ (3)

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Forθv= 1, this theorem also includes the following result of [1].

Corollary 1 Let A= (anv) be an inﬁnite matrix of complex numbers for all n,v ≥ 0 and 1 < k < ∞. Then, A ∈

bv,bvkif and only if(1),(2) hold and

sup ν ∞

∑

n=0 ∞

∑

j=ν an j− an−1, j k < ∞.

Now we give the following theorem.

Theorem 2 Let 1≤ k < ∞ and θ = (θn) be a sequence of positive numbers. If A ∈ bv,bvθ_k, then A_χ= lim_r→∞ ⎛ ⎝

_∑

∞ n=r+1 θ 1−1/k n ∞

∑

j=ν an j− an−1, j k⎞ ⎠ 1/k and A∈ C bvθ_k,bv iff lim r→∞ ∞

∑

n=r+1 θ 1−1/k n ∞

∑

j=ν an j− an−1, j k = 0.

Proof Let T : bv → and T : bvθ_k → k be given by T(x) = xv− xv−1, x−1 = 0 and T(x) =

θ1/k∗

v (xv− xv−1), x−1= 0. Then, it is easy to show that T and T ” are linear bijection preseving norms, i.e.,

xbv= xandxbvθ_k = xk. So, bv and bv θ

k are norm isometric to the spaces and k, respectively, i.e., bv and bvθ_k k. Note that T(x) = y for x ∈ bv. Then, x = T−1(y) ∈ Sbvif and only if y∈ S, where SXis unit sphere. Further, we deﬁne the inﬁnite matrix D by

dnv= θn1−1/k ∞

∑

j=ν an j− an−1, j. Note that we get

bv → bvA θ_k

T↓ T↓

→D k

and so, TAT−1= D and A ∈bv,bvθ_kiff D∈ (,k). Also,

A(bv,bvθ k) = sup_x₀ A(x)_bvθ k xbv = sup_x₀ T−1DT(x)_bvθ k xbv = sup y0 D(y)_k y = D(,k). So, it follows from Lemma 2, Lemma 3 and Lemma 4 that

A_χ = χ (ASbv) = χ(TASbv) = χ(DT_S bv) = lim_r→∞sup y∈S (I − Pr)D(y)_k = lim_r→∞sup y∈S D(r)(y) k= limr→∞ D(r)_(, k) = lim_r→∞sup v _∞ ∑ n=r+1|dnv| k 1/k

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where Pr: → is deﬁned by Pr(y) = (y0,y1,...,yr,0,...), and

dnv(r)=

0, 0 ≤ n ≤ r

dnv, n> r which complete the proof together with Lemma 2.

In the special case, the following result is immediate. Corollary 2 Let 1≤ k < ∞. If A ∈bvk_,bv_{, then}

Aχ= lim_r→∞sup v ⎛ ⎝

_∑

∞ n=r+1 ∞

∑

j=ν an j− an−1, j k⎞ ⎠ 1/k , and A∈ C bv,bvkiff lim r→∞supv ∞

∑

n=r+1 ∞

∑

j=ν an j− an−1, j k = 0.

REFERENCES

[1] E. Malkowsky, V. Rakoˇcevi´c and S. ˇZivkovi´c, Matrix transformations between the sequence space bvkand certain BK spaces,Bull. Cl. Sci. Math. Nat. Sci. Math.123, 33–46 (2002).

[2] G. C. Hazar and M. A. Sarıg¨ol, ”The space bvθ_k and matrix transformations,” in Proceedings of the 8th Internetional Eurasian Conference On Mathematical Sciences And Applications Baku, Azerbaijan, (2019), (in press).

[3] E. Malkowsky and V. Rakoˇcevi´c, An introduction into the theory of sequence space and measures of non-compactness, Zb. Rad. (Beogr) 17, 143-234 (2000).

[4] V. Rakoˇcevi´c, Measures of noncompactness and some applications, Filomat 12, 87-120 (1998).

[5] M. A. Sarıg¨ol, Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci. 42, 3, 28-35 (2015). [6] M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenraumen Eine Ergebnis¨uberischt,Math Z.154,