Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces

Abstract

We study the spaces 𝑤^𝑝₀, 𝑤^𝑝, and 𝑤^𝑝_∞ of sequences that are strongly summable to 0, summable, and bounded with index 𝑝≥1 by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaces 𝑤^𝑝 into the spaces 𝑤¹_∞, 𝑤¹, and 𝑤¹₀. We also give estimates for the operator norm and the Hausdorff measure of noncompactness of such operators. Finally we apply our results to characterize the classes of compact bounded linear operators from 𝑤^𝑝₀ and 𝑤^𝑝 into 𝑤¹₀ and 𝑤¹.

1. Introduction

The spaces 𝑤^𝑝₀, 𝑤^𝑝, and 𝑤^𝑝_∞ of all complex sequences that are strongly summable to zero, strongly summable, and strongly bounded, with index 𝑝≥1 by the Cesàro method of order 1, were first introduced and studied by Maddox [1, 2]. Further recent studies on the spaces 𝑤₀(𝑝) where the constant index 𝑝 in 𝑤^𝑝₀ is replaced by the terms of a positive sequence 𝑝=(𝑝_𝑘) can be found in [3, 4]. Extensive studies of generalizations of the spaces 𝑤^𝑝₀, 𝑤^𝑝, and 𝑤^𝑝_∞ to spaces of sequences of strong weighted means can be found in [5, 6].

In [7], a complete list was given of the characterizations of all matrix transformations from Maddox's spaces into the classical spaces ℓ_∞, 𝑐, 𝑐₀, and ℓ₁ of all bounded, convergent, and null sequences and of all absolutely convergent series. The characterizations of matrix transformations from the classical sequence spaces into Maddox's spaces with index 𝑝=1 were established in [8]. Furthermore, some classes of compact bounded linear operators between those spaces were characterized.

Recently, several authors applied the Hausdorff measure of noncompactness to characterize matrix transformations between sequence spaces that are matrix domains of triangles in the classical sequence spaces, for instance, in [9–14].

In this paper, we extend our studies from the normally considered matrix transformations to the general bounded linear operators from 𝑤^𝑝 into 𝑤¹_∞, 𝑤¹, and 𝑤¹₀. We establish the representations of those operators, deduce estimates for their operator norms and Hausdorff measures of noncompactness, and characterize the corresponding classes of compact bounded linear operators.

2. Notations and Basic Results

In this section we list the notations, concepts, and basic results needed in the paper.

As usual, we denote by 𝜔 and 𝜙 the sets of all complex sequences 𝑥=(𝑥_𝑘)^∞_𝑘=1 and of all sequences that terminate in zeros; also let 𝑒 and 𝑒^(𝑛) for all 𝑛∈ℕ be the sequences with 𝑒_𝑘=1 for all 𝑘 and 𝑒_𝑛^(𝑛)=0 and 𝑒_𝑘^(𝑛)=0 for 𝑘≠𝑛.

A Banach space 𝑋⊂𝜔 is a BK space if each coordinate 𝑃_𝑛∶𝑋→ℂ with 𝑃_𝑛𝑥=𝑥_𝑛 for 𝑥=(𝑥_𝑘)^∞_𝑘=1∈𝑋 is continuous. A BK space 𝑋⊃𝜙 is said to have AK if 𝑥^[𝑚]=∑^𝑚_𝑘=1𝑥_𝑘𝑒^(𝑘)→𝑥(𝑚→∞) for every sequence 𝑥=(𝑥_𝑘)^∞_𝑘=1∈𝑋.

Let (𝑋,‖⋅‖) be a normed space and 𝑆_𝑋={𝑥∈𝑋∶‖𝑥‖=1} and 𝐵_𝑋={𝑥∈𝑋∶‖𝑥‖≤1} denote the unit sphere and closed unit ball in 𝑋, respectively. If 𝑋 and 𝑌 are Banach spaces, then we write ℬ(𝑋,𝑌) for the space of all bounded linear operators 𝐿∶𝑋→𝑌 with the operator norm ‖𝐿‖=sup{‖𝐿(𝑥)‖∶𝑥∈𝑆_𝑋}; we write 𝑋^∗=ℬ(𝑋,ℂ) for the continuous dual of 𝑋, that is, the space of all continuous linear functionals on 𝑋 with the norm ‖𝑓‖=sup{|𝑓(𝑥)|∶𝑥∈𝑆_𝑋}. Furthermore, if (𝑋,‖⋅‖) is a normed sequence space, then we write ‖𝑎‖^∗_𝑋=sup_{𝑥∈𝑆𝑋}|∑^∞_𝑘=1𝑎_𝑘𝑥_𝑘| for 𝑎∈𝜔 provided the expression on the right-hand side exists and is finite which is the case whenever 𝑋 is a BK space and 𝑎∈𝑋^𝛽 [15, Theorem 7.2.9].

For any subset 𝑋 of 𝜔, the set 𝑋^𝛽={𝑎∈𝜔∶∑^∞_𝑘=1𝑎_𝑘𝑥_𝑘convergesforall𝑥∈𝑋} is called the 𝛽-dual of 𝑋.

Let 𝐴=(𝑎_𝑛𝑘)^∞_𝑛,𝑘=1 be an infinite matrix of complex numbers, let 𝑋 and 𝑌 be subsets of 𝜔, and let𝑥∈𝜔. We write 𝐴_𝑛=(𝑎_𝑛𝑘)^∞_𝑘=1 for the sequence in the 𝑛th row of 𝐴, 𝐴_𝑛𝑥=∑^∞_𝑘=1𝑎_𝑛𝑘𝑥_𝑘, 𝐴𝑥=(𝐴_𝑛𝑥)^∞_𝑛=1 (provided all the series converge) and (𝑋,𝑌) for the class of all matrices 𝐴 such that 𝐴_𝑛∈𝑋^𝛽 for all 𝑛∈ℕ and 𝐴𝑥∈𝑌 for all 𝑥∈𝑋. It is known that if 𝑋 and 𝑌 are BK spaces then every matrix 𝐴∈(𝑋,𝑌) defines an operator 𝐿_𝐴∈ℬ(𝑋,𝑌) by 𝐿_𝐴(𝑥)=𝐴𝑥 for all 𝑥∈𝑋 [15, Theorem 4.2.8] and if, in addition, 𝑋 has AK then every operator 𝐿∈ℬ(𝑋,𝑌) is given by a matrix 𝐴∈(𝑋,𝑌) such that 𝐿(𝑥)=𝐴𝑥 for all 𝑥∈𝑋 [16, Theorem 1.9].

Throughout, let 1≤𝑝<∞, and let 𝑞 be the conjugate number of 𝑝; that is, 𝑞=∞ for 𝑝=1 and 𝑞=𝑝/(𝑝−1) for 1<𝑝<∞. We write 𝑤^𝑝₀=𝑥∈𝜔∶lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑥_𝑘^𝑝,𝑤=0_𝑝=𝑥∈𝜔∶𝑥−𝜉⋅𝑒∈𝑤^𝑝₀,𝑤forsome𝜉∈ℂ_𝑝∞=𝑥∈𝜔∶sup_𝑛1𝑛^𝑛_𝑘=1𝑥_𝑘^𝑝<∞ (1) for the sets of all sequences that are strongly summable to 0, strongly summable, and strongly bounded, with index 𝑝 by the Cesàro method of order 1; if 𝑝=1, we write 𝑤₀=𝑤¹₀, 𝑤=𝑤¹, and 𝑤_∞=𝑤¹_∞, for short.

The following results are known and can be found in [1] and, for instance, in [7, Proposition 1.1].

For each sequence 𝑥∈𝑤^𝑝, the 𝑤^𝑝-limit 𝜉 for which 𝑥−𝜉⋅𝑒∈𝑤^𝑝₀ (2) is unique. We write ∑_𝜈=∑^2𝜈+1−1_𝑘=2^𝜈 for 𝜈=0,1,… and max_𝜈=max₂^𝜈_≤𝑘≤2^𝜈+1₋₁.

The sets𝑤^𝑝₀, 𝑤^𝑝, and 𝑤^𝑝_∞ are BK space with the equivalent block and sectional norms ‖𝑥‖_𝑏=sup_𝜈∈ℕ012^𝜈_𝜈𝑥_𝑘^𝑝1/𝑝,‖𝑥‖_𝑠=sup_𝑛1𝑛^𝑛_𝑘=1|𝑥_𝑘|^𝑝1/𝑝. (3)𝑤^𝑝₀ is a closed subspace of 𝑤^𝑝 and 𝑤^𝑝 is a closed subspace of 𝑤^𝑝_∞; 𝑤^𝑝₀ has AK, and every sequence 𝑥=(𝑥_𝑘)^∞_𝑘=1∈𝑤^𝑝 has a unique representation 𝑥=𝜉⋅𝑒+^∞_𝑘=1𝑥_𝑘−𝜉𝑒^(𝑘)where𝜉isthe𝑤^𝑝-limitof𝑥, (4) and 𝑤^𝑝_∞ has no Schauder basis. We always assume that 𝑤^𝑝₀, 𝑤^𝑝, and 𝑤^𝑝_∞ have the block norm, unless explicitly stated otherwise.

We put ‖𝑎‖_ℳ𝑝=^∞_𝜈∑₌₀2^𝜈max_𝜈𝑎_𝑘𝑝=1_∞𝜈∑₌₀2^𝜈/𝑝∑_𝜈|𝑎_𝑘|^𝑞1/𝑞1<𝑝<∞ (5) and ℳ_𝑝={𝑎∈𝜔∶‖𝑎‖_ℳ𝑝<∞}. The following results are known and can be found, for instance, in [7, Proposition 2.1]: 𝑤^𝑝₀^𝛽=𝑤^𝑝𝛽=𝑤^𝑝_∞^𝛽=ℳ_𝑝,𝑤^𝑝₀^∗andℳ_𝑝‖𝑎‖arenormisomorphic,_{∗𝑤𝑝∞}=‖𝑎‖_ℳ𝑝∀𝑎∈𝑤^𝑝_∞; (6)(7)(8)𝑓∈(𝑤^𝑝)^∗ if and only if there are 𝑎₀∈ℂ and a sequence 𝑎=(𝑎_𝑘)^∞_𝑘=1∈ℳ_𝑝 such that 𝑓(𝑥)=𝜉𝑎₀+^∞_𝑘=1𝑎_𝑘𝑥_𝑘∀𝑥∈𝑤^𝑝with𝜉from(2). (9) Moreover 𝑓=𝑎₀+‖𝑎‖_ℳ𝑝∀𝑓∈𝑤^𝑝∗. (10)

We need the following result where we assume that the initial and final spaces have the block and sectional norms, respectively.

Proposition 1. One has the following.(a)𝐴∈(𝑤^𝑝_∞,𝑤_∞)   if and only if‖𝐴‖_(𝑤^𝑝_∞,𝑤∞)=sup_𝑚1𝑚max_{𝑁𝑚⊂{1,2,…,𝑚}𝑛∈𝑁𝑚}𝐴_𝑛ℳ𝑝<∞. (11)Also (𝑤^𝑝_∞,𝑤_∞)=(𝑤^𝑝,𝑤_∞)=(𝑤^𝑝₀,𝑤_∞).(b)𝐴∈(𝑤^𝑝₀,𝑤₀)   if and only if (11) holds andlim_𝑚→∞1𝑚^𝑚_𝑛=1𝑎_𝑛𝑘=0𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑘∈ℕ. (12)(c)𝐴∈(𝑤^𝑝₀,𝑤)    if and only if (11) holds and𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑘∈ℕ𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝛼_𝑘∈ℂ𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑎_𝑛𝑘−𝛼_𝑘=0. (13)(d)𝐴∈(𝑤^𝑝,𝑤₀)   if and only if (11) and (12) hold andlim_𝑚→∞1𝑚^𝑚_𝑛=1^∞_𝑘=1𝑎_𝑛𝑘=0. (14)(e)𝐴∈(𝑤^𝑝,𝑤)   if and only if (11) and (13) hold andlim_𝑚→∞1𝑚^𝑚_𝑛=1^∞_𝑘=1𝑎_𝑛𝑘−𝛼=0𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈ℂ. (15)(f)If 𝐴∈(𝑋,𝑌)   in the cases above, then‖𝐴‖_(𝑤^𝑝_∞,𝑤∞)≤𝐿_𝐴≤4⋅‖𝐴‖_(𝑤^𝑝_∞,𝑤∞). (16)

Proof. (a) follows from [17, Corollary 1], (6), and (7).
(b) and (c) follow from [15, 8.3.6, p.   123], since 𝑤^𝑝₀ and 𝑤^𝑝 are closed subspaces of 𝑤^𝑝_∞.
(d) and (e) follow from Parts (b) and (c) by [15, 8.3.7].
(f) follows from [17, equation (2.8)], (6), and (7).

3. Representation of Bounded Linear Operators

Here we establish the representations of the bounded linear operators in ℬ(𝑤^𝑝,𝑌) for 𝑌=𝑤_∞,𝑤,𝑤₀ and give estimates for the operator norms in each case. Throughout, we assume that 𝑤^𝑝 and 𝑌 have the block and sectional norms, respectively.

We note that, since 𝑤^𝑝₀ has AK, every 𝐿∈ℬ(𝑤^𝑝₀,𝑌) is given by an infinite matrix 𝐴∈(𝑤^𝑝₀,𝑌), and its operator norm satisfies the inequalities in (16).

Theorem 2. (a) One has 𝐿∈ℬ(𝑤^𝑝,𝑤_∞) if and only if there exist a matrix 𝐴∈(𝑤^𝑝₀,𝑤_∞) and a sequence 𝑏∈𝑤_∞ such that 𝐿(𝑥)=𝑏⋅𝜉+𝐴𝑥∀𝑥∈𝑤^𝑝,𝑤ℎ𝑒𝑟𝑒𝜉𝑖𝑠𝑡ℎ𝑒𝑤^𝑝-𝑙𝑖𝑚𝑖𝑡𝑜𝑓𝑥. (17) Moreover, one has sup_𝑚1𝑚max_{𝑁𝑚⊂{1,…,𝑚}𝑛∈𝑁𝑚}𝑏_𝑛+_{𝑛∈𝑁𝑚}𝐴_𝑛ℳ𝑝≤‖𝐿‖≤4⋅sup_𝑚1𝑚max_{𝑁𝑚⊂{1,…,𝑚}𝑛∈𝑁𝑚}𝑏_𝑛+_{𝑛∈𝑁𝑚}𝐴_𝑛ℳ𝑝. (18)
(b) One has 𝐿∈ℬ(𝑤^𝑝,𝑤) if and only if there exist a matrix 𝐴∈(𝑤^𝑝₀,𝑤) and a sequence 𝑏∈𝑤_∞ with lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑏_𝑛+𝐴_𝑛𝑒−𝛽=0𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛽∈ℂ (19) such that (17) holds; moreover, one has (18).
(c) One has 𝐿∈ℬ(𝑤^𝑝,𝑤₀) if and only if there exist a matrix 𝐴∈(𝑤^𝑝₀,𝑤₀) and a sequence 𝑏∈𝑤_∞ with lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑏_𝑛+𝐴_𝑛𝑒=0 (20) such that (17) holds; moreover, one has (18).

Proof. (a) First we assume 𝐿∈ℬ(𝑤^𝑝,𝑤_∞) and write 𝐿_𝑛=𝑃_𝑛∘𝐿 for 𝑛=1,2,… where 𝑃_𝑛 denotes the 𝑛th coordinate. Since 𝑤^𝑝 is a 𝐵𝐾 space, it follows that 𝐿_𝑛∈(𝑤^𝑝)^∗ for each 𝑛, and hence we have by (9) 𝐿_𝑛(𝑥)=𝑏_𝑛⋅𝜉+𝐴_𝑛𝑥∀𝑥∈𝑤^𝑝,where𝜉isthe𝑤^𝑝𝑏-limitof𝑥,_𝑛=𝐿_𝑛(𝑒)−^∞_𝑘=1𝐿_𝑛𝑒^(𝑘),𝑎_𝑛𝑘=𝐿_𝑛𝑒^(𝑘)𝐴∀𝑛and𝑘,_𝑛∈ℳ_𝑝∀𝑛. (21)(22) This yields (17); moreover, we have by (10) 𝐿_𝑛=𝑏_𝑛+𝐴_𝑛ℳ𝑝for𝑛=1,2,…. (23) It also follows from (21) and 𝐿(𝑒)∈𝑤_∞ that sup_𝑚1𝑚^𝑚_𝑛=1𝐿_𝑛(𝑒)=sup_𝑚1𝑚^𝑚_𝑛=1𝑏_𝑛+𝐴𝑒<∞, (24) and so 𝑏+𝐴𝑒∈𝑤_∞. Furthermore, since 𝐿(𝑥⁽⁰⁾)=𝐴𝑥⁽⁰⁾ for all 𝑥⁽⁰⁾∈𝑤^𝑝₀, we have 𝐴∈(𝑤^𝑝₀,𝑤_∞)=(𝑤^𝑝_∞,𝑤_∞), and so 𝐴𝑒∈𝑤_∞ and we obtain 𝑏=(𝑏+𝐴𝑒)−𝐴𝑒∈𝑤_∞.
Now we show (18). We define 𝐿_𝑁𝑚∶𝑤^𝑝→ℂ for all 𝑚∈ℕ and for each subset 𝑁_𝑚 of {1,2,…,𝑚} by 𝐿_𝑁𝑚=1𝑚_{𝑚∈𝑁𝑚}𝐿_𝑛. (25) Then clearly 𝐿_𝑁𝑚∈(𝑤^𝑝)^∗, and we obtain by a well-known inequality (cf. [18]) 𝐿_𝑁𝑚(𝑥≤1)𝑚^𝑚_𝑛=1𝐿_𝑛(𝑥)≤4⋅max_{𝑁𝑚⊂{1,…,𝑚}}𝐿_𝑁𝑚(𝑥), (26) and hence by the first inequality in (26) and by (10) 𝐿_𝑁𝑚=1𝑚_{𝑛∈𝑁𝑚}𝐿_𝑛=1𝑚_{𝑛∈𝑁𝑚}𝑏_𝑛+_{𝑛∈𝑁𝑚}𝐴_𝑛ℳ𝑝≤‖𝐿‖ (27) for all 𝑚∈ℕ and all 𝑁_𝑚⊂{1,2,…,𝑚}, and so the first inequality in (18) follows. Also, we obtain for all 𝑚∈ℕ from the second inequality in (26) and by (10) 1𝑚^𝑚_𝑛=1𝐿_𝑛(𝑥)≤4⋅max_{𝑁𝑚⊂{}1,…,𝑚}𝐿_𝑁𝑚≤4⋅sup_𝑚1𝑚max_{𝑁𝑚⊂{1,…,𝑚}𝑛∈𝑁𝑚}𝑏_𝑛+_{𝑛∈𝑁𝑚}𝐴_𝑛ℳ𝑝. (28) This implies the second inequality in (18).
Conversely, we assume that 𝐴∈(𝑤^𝑝₀,𝑤_∞), that 𝑏∈𝑤_∞, and that (17) is satisfied. Let 𝜀>0, and𝑥∈𝑤^𝑝 be given and let 𝜉 be the 𝑤^𝑝-limit of 𝜉. Then there exists 𝑀∈ℕ such that (1/𝑚)∑^∞_𝑛=1|𝑥_𝑘−𝜉|^𝑝<𝜀^𝑝 for all 𝑚>𝑀. Thus we have for all 𝑚>𝑀𝜉=1𝑚^𝑚_𝑛=1|𝜉|^𝑝1/𝑝≤1𝑚^𝑚_𝑛=1|𝑥_𝑛|^𝑝1/𝑝+1𝑚^𝑚_𝑛=1|𝑥_𝑛−𝜉|^𝑝1/𝑝≤‖𝑥‖_𝑏+𝜀. (29) Since 𝜀 was arbitrary, we have 𝜉≤‖𝑥‖_𝑏. (30) We define the map 𝑔∶𝑤^𝑝→𝑤_∞ with 𝑔(𝑥)=𝑏𝜉 for all 𝑥∈𝑤^𝑝, where 𝜉∈ℂ is the 𝑤^𝑝-limit of 𝑥. Then 𝑔 trivially is linear, and it follows from (30) that 𝑔(𝑥)_𝑏=sup_𝑚1𝑚^𝑚_𝑛=1𝑏_𝑛𝜉≤sup_𝑚1𝑚^𝑚_𝑛=1𝑏_𝑛⋅‖𝑥‖_𝑏, (31) and, since 𝑏∈𝑤_∞, we obtain 𝑔∈ℬ(𝑤^𝑝,𝑤_∞). Furthermore, we have 𝐴∈(𝑤^𝑝₀,𝑤_∞)=(𝑤^𝑝,𝑤_∞), and hence 𝐿_𝐴∈ℬ(𝑤^𝑝,𝑤_∞), and so, by (17), 𝐿=𝑔+𝐿_𝐴∈ℬ(𝑤^𝑝,𝑤_∞).
(b) First we assume 𝐿∈ℬ(𝑤^𝑝,𝑤). Then 𝐿∈ℬ(𝑤^𝑝,𝑤_∞), and by Part (a) there are 𝑏∈𝑤_∞ and 𝐴∈(𝑤^𝑝₀,𝑤_∞)=(𝑤^𝑝,𝑤_∞) such that (17) is satisfied; also clearly (18) is satisfied. It follows from (17), 𝐿(𝑒)∈𝑤, and 𝐿(𝑒^(𝑘))∈𝑤 for each 𝑘 that there exist 𝛽∈ℂ and 𝛼_𝑘∈ℂ for 𝑘=1,2,… such that lim_𝑚→∞1𝑚^𝑚_𝑛=1𝐿_𝑛(𝑒)−𝛽=lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑏_𝑛+𝐴_𝑛𝑒−𝛽=0, (32) which is (19), and lim_𝑚→∞1𝑚^𝑚_𝑛=1𝐿_𝑛𝑒^(𝑘)−𝛼_𝑘=lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑎_𝑛𝑘−𝛼_𝑘=0foreach𝑘. (33) Now it follows from 𝐴∈(𝑤^𝑝₀,𝑤_∞) and (33) by Proposition 1(c) that 𝐴∈(𝑤^𝑝₀,𝑤).
Conversely, we assume that 𝐴∈(𝑤^𝑝₀,𝑤), that 𝑏∈𝑤_∞, and that (17) and (19) are satisfied. Then we have 𝐴∈(𝑤^𝑝₀,𝑤_∞), and so 𝐿∈ℬ(𝑤^𝑝,𝑤_∞) by Part (a). Let 𝑥∈𝑤^𝑝 be given and let 𝜉 be the 𝑤^𝑝-limit of 𝑥. Then we have 𝑥⁽⁰⁾=𝑥−𝜉𝑒∈𝑤^𝑝₀ and, by (17), 𝐿_𝑛(𝑥)=𝑏_𝑛𝜉+^∞_𝑘=1𝑎_𝑛𝑘𝑥_𝑘=𝑏_𝑛+𝐴_𝑛𝑒𝜉+𝐴_𝑛𝑥⁽⁰⁾. (34) Since 𝐴∈(𝑤^𝑝₀,𝑤), the 𝑤-limit of 𝜂₀ of 𝐴𝑥⁽⁰⁾ exists and we have by (19) and (34) 0≤lim_𝑚→∞1𝑚^𝑚_𝑛=1𝐿_𝑛(𝑥)−𝛽𝜉+𝜂₀≤lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑏_𝑛+𝐴_𝑛𝑒−𝛽⋅𝜉+lim_𝑚→∞1𝑚^𝑚_𝑛=1𝐴_𝑛𝑥⁽⁰⁾−𝜂₀=𝜉⋅lim_𝑚→∞1𝑚^𝑚_𝑛=1𝑏_𝑛+𝐴_𝑛𝑒−𝛽=0. (35) Therefore we have 𝐿∈ℬ(𝑤^𝑝,𝑤).
(c) The proof of Part (c) is similar to that of Part (b) with 𝛽=0 and 𝛼_𝑘=0(𝑘=1,2,…).

Remark 3. It was shown in the proof of [19, Theorem 3.6] that if 𝐴∈(𝑤^𝑝₀,𝑤) then (𝛼_𝑘)^∞_𝑘=1∈ℳ_𝑝 with 𝛼_𝑘(𝑘=1,2,…) from (33) and in [19, equation (3.14)] that the 𝑤-limit of 𝐴𝑥⁽⁰⁾ for any sequence 𝑥⁽⁰⁾ in 𝑤^𝑝₀ is given by 𝜂₀=^∞_𝑘=1𝛼_𝑘𝑥_𝑘⁽⁰⁾. (36) Let 𝐿∈ℬ(𝑤^𝑝,𝑤) and 𝑥∈𝑤^𝑝 and let 𝜉 be the 𝑤^𝑝-limit of 𝑥; then we obtain by (35) and (36) for the 𝑤-limit of 𝐿(𝑥)𝜂=𝛽⋅𝜉+𝜂₀=𝛽⋅𝜉+^∞_𝑘=1𝛼_𝑘𝑥_𝑘=−𝜉𝛽−^∞_𝑘=1𝛼_𝑘⋅𝜉+^∞_𝑘=1𝛼_𝑘𝑥_𝑘with𝛽from(19). (37)

4. Compact Operators

In this section, we establish estimates for the Hausdorff measures of noncompactness of linear operators and characterize some classes of compact operators from 𝑋 into 𝑌, where 𝑋=𝑤^𝑝₀,𝑤^𝑝 and 𝑌=𝑤₀,𝑤.

First we recall some useful definitions and results. The reader is referred to the monographs [20–23] for the theory and applications of measures of noncompactness. Let 𝑋 and 𝑌 be Banach spaces and let 𝐿∶𝑋→𝑌 be a linear operator. Then 𝐿 is said to be compact if its domain is all of 𝑋 and, for every bounded sequence (𝑥_𝑛)^∞_𝑛=1 in 𝑋, the sequence (𝐿(𝑥_𝑛))^∞_𝑛=1 has a convergent subsequence in 𝑌. We denote the class of such operators by 𝒞(𝑋,𝑌).

Let (𝑋,𝑑) be a metric space, 𝐵(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑦,𝑥)<𝑟} denote the open ball of radius 𝑟 and centre in 𝑥, and ℳ_𝑋 denote the class of bounded subsets of 𝑀. Then the map 𝜒∶ℳ_𝑋→[0,∞) with 𝜒(𝑄)=inf𝜀>0∶𝑄⊂^𝑛_𝑘=1𝐵𝑥_𝑘,𝑟_𝑘,𝑥_𝑘∈𝑋,𝑟_𝑘<𝜀(𝑘=1,…,𝑛;𝑛∈ℕ) (38) is called the Hausdorff measure of noncompactness.

Let 𝑋 and 𝑌 be Banach spaces and let 𝜒₁ and 𝜒₂ be the Hausdorff measures of noncompactness on 𝑋 and 𝑌. Then the operator 𝐿∶𝑋→𝑌 is said to be (𝜒₁,𝜒₂)-bounded if 𝐿(𝑄)∈ℳ_𝑌 for every 𝑄∈ℳ_𝑋, and there exists a positive constant 𝐶 such that 𝜒₂(𝐿(𝑄))≤𝐶⋅𝜒₁(𝑄) for every 𝑄∈ℳ_𝑋. If an operator 𝐿 is (𝜒₁,𝜒₂)-bounded, then ‖𝐿‖_{(𝜒1,𝜒2)}=inf𝐶>0∶𝜒₂(𝐿(𝑄))≤𝐶⋅𝜒₁(𝑄)∀𝑄∈ℳ_𝑋 (39) is called the (𝜒₁,𝜒₂)-measure of noncompactness of 𝐿. In particular, if 𝜒₁=𝜒₂=𝜒, then we write ‖𝐿‖_𝜒=‖𝐿‖_(𝜒,𝜒).

The following useful results are well known.

Proposition 4. Let 𝑋 and 𝑌 be Banach spaces and 𝐿∈ℬ(𝑋,𝑌). Then one has ‖𝐿‖_𝜒=𝜒𝐿𝐵_𝑋=𝜒𝐿𝑆_𝑋 (40) (see [23, Theorem 2.25]), 𝐿∈𝒞(𝑋,𝑌)𝑖𝑓𝑓‖𝐿‖_𝜒=0 (41) (see [23, Corollary 2.26, equation (2.58)]).

We also need the following results which are an immediate consequence of [8, Proposition 3.2 and Lemma 3.5].

Proposition 5. (a) Let 𝒫_𝑛∶𝑤→𝑤 be the projectors onto the linear span of {𝑒,𝑒⁽¹⁾,…,𝑒^(𝑛)}, 𝐼∶𝑤→𝑤 the identity, and ℛ_𝑛=𝐼−𝒫_𝑛 for 𝑛=0,1,…. Then one has for all 𝑄∈ℳ_𝑤12⋅lim_𝑛→∞sup_𝑥∈𝑄ℛ_𝑛(𝑥)≤𝜒(𝑄)≤lim_𝑛→∞sup_𝑥∈𝑄ℛ_𝑛(𝑥). (42)
(b) Let 𝒫_𝑛∶𝑤₀→𝑤₀ be the projectors onto the linear span of {𝑒⁽¹⁾,𝑒⁽²⁾,…,𝑒^(𝑛)}, 𝐼∶𝑤₀→𝑤₀ the identity, and ℛ_𝑛=𝐼−𝒫_𝑛 for 𝑛=0,1,…. Then one has for all 𝑄∈ℳ_𝑤0𝜒(𝑄)=lim_𝑛→∞sup_𝑥∈𝑄ℛ_𝑛(𝑥). (43)

Proof. (a) The inequalities in (42) follow from [8, Proposition 3.2 and Lemma 3.5].
(b) The identity in (42) follows from [8, Proposition 3.2 and Lemma 3.3(a)].

Now we give estimates for the Hausdorff measures of noncompactness of the general operators 𝐿∈ℬ(𝑤^𝑝,𝑤) and 𝐿∈ℬ(𝑤^𝑝,𝑤₀). Let 𝑚,𝑟∈ℕ and 𝑚>𝑟. Then we write 𝑁(𝑚,𝑟) for any subset of the set {𝑟+1,𝑟+2,…,𝑚}.

Theorem 6. (a) Let 𝐿∈ℬ(𝑤^𝑝,𝑤). One uses the notations of Theorem 2 and writes 𝛾_𝑛=𝑏_𝑛−𝛽+∑^∞_𝑘=1𝛼_𝑘 for 𝑛=1,2,… and 𝐶=(𝑐_𝑛𝑘)^∞_𝑛,𝑘=1 for the matrix with 𝑐_𝑛𝑘=𝑎_𝑛𝑘−𝛼_𝑘 for all 𝑛 and 𝑘. Then one has 12lim_𝑟→∞sup_𝑚1𝑚max_{𝑁(𝑚,𝑟)𝑛∈𝑁(𝑚,𝑟)}𝛾_𝑛+_{𝑛∈𝑁(𝑚,𝑟)}𝐶_𝑛ℳ𝑝≤‖𝐿‖_𝜒≤4⋅lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝛾_𝑛+_{()𝑛∈𝑁𝑚,𝑟}𝐶_𝑛ℳ𝑝. (44)
(b) Let 𝐿∈ℬ(𝑤^𝑝,𝑤₀). Then one has lim_𝑟→∞sup_𝑚1𝑚max_{𝑁(𝑚,𝑟)𝑛∈𝑁(𝑚,𝑟)}𝑏_𝑛+_{𝑛∈𝑁(𝑚,𝑟)}𝐴_𝑛ℳ𝑝≤‖𝐿‖_𝜒≤4⋅lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝑏_𝑛+_{𝑛∈𝑁(𝑚,𝑟)}𝐴_𝑛ℳ𝑝. (45)

Proof. (a) We assume 𝐿∈ℬ(𝑤^𝑝,𝑤).
Let 𝑥∈𝑤^𝑝 be given, 𝜉 be the 𝑤^𝑝-limit of 𝑥, and 𝑦=𝐿(𝑥). Then we have from Theorem 2(b) and (17) that 𝑦=𝑏⋅𝜉+𝐴𝑥, where 𝐴∈(𝑤^𝑝₀,𝑤) and 𝑏∈𝑤_∞, and it follows that 𝑦_𝑛=𝑏_𝑛⋅𝜉+𝐴_𝑛𝑥=𝑏_𝑛+^∞_𝑘=1𝑎_𝑛𝑘⋅𝜉+𝐴_𝑛(𝑥−𝜉⋅𝑒)∀𝑛∈ℕ. (46) Furthermore, the complex numbers 𝛽 and 𝛼_𝑘(𝑘=1,2,…) satisfying (19) and (33) exist by Theorem 2(b), and (𝛼_𝑘)^∞_𝑘=1∈ℳ_𝑝 by Remark 3, so 𝜂∈ℂ defined in (37) is the 𝑤-limit of 𝑦=𝐿(𝑥) by Remark 3. Now let 𝒫_𝑟∶𝑤→𝑤 be the projector onto the linear span of {𝑒,𝑒⁽¹⁾,…,𝑒^(𝑟)} and ℛ_𝑟=𝐼−𝒫_𝑟 for 𝑟=0,1,…, where 𝐼 is the identity on 𝑤; hence ℛ_𝑟(𝑦)=∑^∞_𝑛=𝑟+1(𝑦_𝑛−𝜂)𝑒^(𝑛) by (4). Let 𝑟∈ℕ be given. We write 𝑓_𝑛,𝑟(𝑥)=(ℛ_𝑟(𝐿(𝑥)))_𝑛 for all 𝑛 and obtain 𝑓_𝑛,𝑟(𝑥)=0 for 𝑛≤𝑟 and, for 𝑛>𝑟, from (46) and (37) 𝑓_𝑛,𝑟(𝑥)=𝑦_𝑛−𝜂=𝑏_𝑛⋅𝜉+𝐴_𝑛𝑥−𝛽⋅𝜉+^∞_𝑘=1𝛼_𝑘𝑥_𝑘=𝑏−𝜉_𝑛−𝛽+^∞_𝑘=1𝛼_𝑘⋅𝜉+^∞_𝑘=1𝑎_𝑛𝑘−𝛼_𝑘𝑥_𝑘=𝛾_𝑛⋅𝜉+𝐶_𝑛𝑥. (47) Since 𝑓_𝑛,𝑟∈(𝑤^𝑝)^∗, we obtain by the same kind of argument as in the proof of (18) that sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝛾_𝑛+_{()𝑛∈𝑁𝑚,𝑟}𝐶_𝑛ℳ𝑝≤sup_{𝑥∈𝑆𝑤𝑝}ℛ_𝑟(𝐿(𝑥))≤4⋅sup_𝑚1𝑚max_{𝑁(𝑚,𝑟)𝑛∈𝑁(𝑚,𝑟)}𝛾_𝑛+_{𝑛∈𝑁(𝑚,𝑟)}𝐶_𝑛ℳ𝑝. (48) Now the inequalities in (44) follow from (40) and (42).
(b) Now 𝛽=𝛼_𝑘=0 for 𝑘=1,2,…, and as in the proof of Part (a) we obtain (48) with 𝛾_𝑛 and 𝐶_𝑛 replaced by 𝑏_𝑛 and 𝐴_𝑛, respectively, and the inequalities in (45) follow from (40) and (43).

Corollary 7. (a) Let 𝐿∈ℬ(𝑤^𝑝₀,𝑤). Then 𝐿 is given by a matrix 𝐴∈(𝑤^𝑝₀,𝑤) and one has, writing 𝐶=(𝑐_𝑛𝑘)^∞_𝑛,𝑘=1 for the matrix with 𝑐_𝑛𝑘=𝑎_𝑛𝑘−𝛼_𝑘 for all 𝑛 and 𝑘, where 𝛼_𝑘 is given by (33), 12lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝐶_𝑛ℳ𝑝≤‖𝐿‖_𝜒≤4⋅lim_𝑟→∞sup_𝑚1𝑚max_{𝑁(𝑚,𝑟)𝑛∈𝑁(𝑚,𝑟)}𝐶_𝑛ℳ𝑝. (49)
(b) Let 𝐿∈ℬ(𝑤^𝑝₀,𝑤₀). Then one has lim_𝑟→∞sup_𝑚1𝑚max_{𝑁(𝑚,𝑟)𝑛∈𝑁(𝑚,𝑟)}𝐴_𝑛ℳ𝑝≤‖𝐿‖_𝜒≤4⋅lim_𝑟→∞sup_𝑚1𝑚max_{𝑁(𝑚,𝑟)𝑛∈𝑁(𝑚,𝑟)}𝐴_𝑛ℳ𝑝. (50)

Proof. (a) The estimates in (49) are easily obtained from [8, Theorem 3.6] with ‖⋅‖^∗_{(𝑋,𝑤∞)}=‖⋅‖_(𝑤^𝑝_∞,𝑤∞) defined in (11).
(b) The estimates in (50) follow from those in (45) with 𝑏_𝑛=0 for all 𝑛.

We apply our results and close with the characterizations of the classes 𝒞(𝑋,𝑌) for 𝑋=𝑤^𝑝,𝑤^𝑝₀ and 𝑌=𝑤,𝑤₀.

Corollary 8. Let 𝐿∈ℬ(𝑋,𝑌). Then the necessary and sufficient conditions for 𝐿∈𝒞(𝑋,𝑌) can be read from Table 1 where (1)lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝐴_𝑛ℳ𝑝(2)=0,lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝑏_𝑛+_{()𝑛∈𝑁𝑚,𝑟}𝐴_𝑛ℳ𝑝(3)=0,lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝐶_𝑛ℳ𝑝(4)=0,lim_𝑟→∞sup_𝑚1𝑚max_{𝑁()𝑚,𝑟()𝑛∈𝑁𝑚,𝑟}𝛾_𝑛+_{()𝑛∈𝑁𝑚,𝑟}𝐶_𝑛ℳ𝑝=0. (51)

Table 1 

Proof. The conditions in (1)–(4) are immediate consequences of (41) and the conditions in (50), (45), (49), and (44), respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.