Measure of noncompactness of operators in Banach spaces

Abstract

This paper is committed to dealing with the measure of noncompactness of operators in Banach spaces. First, we give a characterization of the measure of noncompact Hausdorff operators with respect to the Hausdorff metric. Then, we show a formula of the Hausdorff measure of noncompactness of operators in . Finally, several common equivalent measures of noncompactness of operators and related proofs are provided.

Keywords

Measure of noncompactness / measure of noncompactness of operators / equivalent measure / Banach space

1 Introduction

The measure of noncompactness was first proposed by Kuratowski [17] in 1930, and later it was named the measure of noncompactness of a set or the Kuratowski measure of noncompactness (denoted as α): Let X be a metric space, and Q be a nonempty bounded set in X. Then

\alpha \left(Q\right)=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\epsilon >0:Q\subset \bigcup _{i=1}^n S_i,\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left(S_i\right)\le \epsilon ,S_i\subset X,i=1,\dots ,n,n\in \mathbb{N}\right\}.

Let \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left(Q\right)=\mathrm{s}\mathrm{u}\mathrm{p}\{d\left(x,y\right)\mid x,y\in Q\}, and there is \alpha \left(Q\right)\le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left(Q\right). α can be used to measure how far a nonempty bounded set in the metric space X is “away” from a compact set, and it has the following properties (where A, B represent any nonempty bounded sets in X):

a) \alpha \left(A\right)=0\iff A is a relatively compact set;

b) \alpha \left(\overline{A}\right)=\alpha \left(A\right);

c) A\subset B\Rightarrow \alpha \left(A\right)\le \alpha \left(B\right);

d) \alpha (A\cup B)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\alpha \right(A),\alpha (B\left)\right\};

Furthermore, when X is a normed space, α also satisfies

e) \alpha \left(kA\right)=\left|k\right|\alpha \left(A\right), where kA=\{x\mid x=ka,a\in A\},k\in F;

f) \alpha \left(A+B\right)\le \alpha \left(A\right)+\alpha \left(B\right), where A+B=\{x=a+b,a\in A,b\in B\};

g) \alpha (\mathrm{c}\mathrm{o}A)=\alpha \left(A\right);

h) \alpha (A+x_0)=\alpha \left(A\right),\mathrm{\forall }{x}_0\in X.

In addition, α is continuous. That is, for any nonempty bounded sets A, B in X, given any \epsilon >0,\mathrm{\exists }\delta >0, when, there is . These properties of \alpha are almost deduced in parallel from the properties of the diameter (diam). \alpha has the following property.

Lemma 1.1 [11, 23]　 Let X be a normed linear space and B_X be the closed unit ball of X. When X is finite dimensional, \alpha \left(B_X\right)=0; when X is infinite dimensional, \alpha \left(B_X\right)=2.

In 1957, Gohberg et al. [13] introduced the Hausdorff measure of noncompactness β (also known as the ball measure of noncompactness): Let X be a real Banach space and Q be a nonempty bounded set in X. Then

which is equivalent to

\beta \left(Q\right)=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\epsilon >0:Q\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\epsilon -\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{n}X\right\},

which is equivalent to

\beta \left(Q\right)=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\epsilon >0:Q\subset K+\epsilon B_X\right\},

where K is a compact set in X.

Subsequently, Goldstein and Markus [14], Sadovskiĭ [24], Goebel [12], et al. further studied the Hausdorff measure of noncompactness and proved that β also satisfied the above-mentioned properties a)‒h). β also had the following property.

Lemma 1.2 [6]　 Let X be a normed linear space. When X is finite dimensional, \beta \left(B_X\right)=0; when X is infinite dimensional, \beta \left(B_X\right)=1.

In 1955, Darbo [8] first defined the Darbo function T by the measure of noncompactness \alpha of a set and proved that every continuous self-mapping T on a bounded closed convex set had a fixed point. Darbo's fixed-point theorem is an important generalization of the Banach fixed-point theorem and the Schauder fixed-point theorem. Since the measure of noncompactness of an operator has extensive applications in fields such as fixed points, differential equations, and integral equations (see [1, 3-5], etc.), applications in fractional order partial differential equations can be found in [10], and the characterization of compact operators in Banach spaces can be seen in [9, 20], etc. Therefore, the study of the ball measure of noncompactness of operators on Banach spaces is of particular importance. Reference [7] studied non-equivalent measures of noncompactness in Banach spaces. Inspired by [7], for a measure of noncompactness μ, if we only consider the measure of this type of bounded sets \left\{T{B}_X\right\}, where \mathrm{\forall }T\in B\left(X\right), and B(X) represents the space of bounded linear operators on X, then \mu \left(T\right)=\mu \left(T{B}_X\right) is the measure of noncompactness of the operator T. In this paper, some conclusions about the measure of noncompactness of operators in Banach spaces are obtained.

2 Measures of noncompactness of operators and the Hausdorff metric

The following shows a characterization of the measure of noncompactness of the Hausdorff operator in terms of the Hausdorff metric.

Let (X, d) be a metric space. Denoted by \mathcal{B}\left(\mathrm{X}\right) the family of all nonempty, bounded and closed sets on X. Let A,B\in \mathcal{B}\left(X\right), the Hausdorff distance d_H between A and B is defined as

\begin{array}{rl}{d}_H\left(A,B\right)& =\mathrm{m}\mathrm{a}\mathrm{x}\left\{\underset{a\in A}{\mathrm{s}\mathrm{u}\mathrm{p}}d\left(a,B\right),\underset{b\in B}{\mathrm{s}\mathrm{u}\mathrm{p}}d\left(A,b\right)\right\}\\ & =\mathrm{m}\mathrm{a}\mathrm{x}\left\{\underset{a\in A}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{b\in B}{\mathrm{i}\mathrm{n}\mathrm{f}}\Vert a-b\Vert ,\underset{b\in B}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{a\in A}{\mathrm{i}\mathrm{n}\mathrm{f}}\Vert a-b\Vert \right\}.\end{array}

If \varnothing \ne F\subset X,r>0, denote

B(F,r)=\bigcup _{x\in F} B(x,r)=\{y\in X:d(y,F)\le r\},

then there is

d_H\left(A,B\right)=\mathrm{i}\mathrm{n}\mathrm{f}\{r>0:A\subset B\left(B,r\right),B\subset B\left(A,r\right)\}.

It can be seen from the above definitions that the Hausdorff distance d_H is a metric. \mathcal{B}\left(\mathrm{X}\right) is endowed with the Hausdorff metric. According to [21], if X is a complete metric space, then \mathcal{B}\left(\mathrm{X}\right) is also complete, and \left(\mathcal{K}\left(X\right),d_H\right) is a closed subspace of \mathcal{B}\left(\mathrm{X}\right). Denote β as the ball measure of noncompactness, B_X as the unit ball of X, then \beta \left(T\right)=\beta \left(T{B}_X\right)is called the ball operator measure of noncompactness.

Theorem 2.1　 Let X be a Banach space, B_X be the unit ball of X, B(X) be the set of all bounded linear operators on X, \mathrm{\forall }T,T_1,T_2\in B\left(X\right), and denote K as the family of all nonempty compact sets in X. Then the following holds:

\begin{array}{rl}\left|\beta \right(T_1)-\beta (T_2\left)\right|& \le d_H\left(T_1\right(B_X),T_2(B_X)),\\ \beta \left(T\right)& =d_H(T{B}_X,K).\end{array}

Proof　Let \mathrm{\forall }r>0,d=d_H\left(T_1{B}_X,T_2{B}_X\right). Then there exists a finite set S\subset X, such that

(2.1)T_1{B}_X\subset B\left(T_2{B}_X,d+r\right),T_2{B}_X\subset B\left(S,\beta \left(T_2{B}_X\right)+r\right).

According to (2.1), there is

(2.2)T_1{B}_X\subset B\left(S,\beta \left(T_2{B}_X\right)+d+2r\right).

So,

(2.3)\beta \left(T_1\right)\le \beta \left(T_2\right)+d+2r.

By interchanging A, B, the first equation of the conclusion holds.

It is known that \beta \left(T\right)\le d_H(T{B}_X,\mathcal{K}) from \left|\beta \right(T_1)-\beta (T_2\left)\right|\le d_H\left(T_1\right(B_X),T_2(B_X)). On the other hand, for ∀r > 0, there exists a finite set F\subset X, such that

(2.4)T{B}_X\subset B\left(F,\beta \left(T{B}_X\right)+r\right),F\subset B\left(T{B}_X,\beta \left(T\right)+r\right).

d_H(T{B}_X,\mathcal{K})\le d_H(T{B}_X,F)\le \beta \left(T\right)+r, so, the conclusion holds.

Definition 2.1 [7]　Let G be an abelian semigroup and the number field \mathbb{F}\in \{\mathbb{R},\mathbb{C}\}. If the operators \left(x,y\right)\in G\times G\to x+y\in G and \left(\alpha ,x\right)\in \mathbb{F}\times G\to \alpha x\in G satisfy \mathrm{\forall }\lambda, \mu \in \mathbb{F} and \mathrm{\forall }g,g_1,g_2\in G, the following conditions are established:

\begin{array}{rl}\left(\lambda \mu \right)g& =\lambda \left(\mu g\right),\\ \lambda \left(g_1+g_2\right)& =\lambda g_1+\lambda g_2,\\ 1g& =g,\\ 0g& =0.\end{array}

Then G is called a module. If a norm is assigned to the module G, then G is called a normed semigroup.

Let X be a Banach space. Define the following addition and scalar multiplication on B(X):

A\oplus B=\overline{A+B}=\overline{\{a+b:a\in A,b\in B\}},

\lambda \cdot A=\{\lambda a:a\in A\},

where \mathrm{\forall }A,B\in \mathcal{B}\left(X\right),\lambda \in \mathbb{F}, and from the definition it is known that \left(B\left(X\right),\oplus ,\right) is a module, and \mathcal{K}\left(X\right) is also a module.

\mathrm{\forall }A+\mathcal{K}\left(X\right)\in \mathcal{B}\left(X\right)/\mathcal{K}\left(X\right), denote \left[A\right]=A+\mathcal{K}\left(X\right). The addition and scalar multiplication operations inherited in the quotient semigroup \mathcal{B}\left(X\right)/\mathcal{K}\left(X\right) are as follows: \mathrm{\forall }\left[A\right],\left[B\right]\in \frac{\mathcal{B}\left(X\right)}{\mathcal{K}\left(X\right)},\mathrm{\forall }\lambda \in \mathbb{F},

\begin{array}{rl}\left[A\right]+\left[B\right]& =[A+B],\\ \lambda \left[A\right]& =\left[\lambda A\right].\end{array}

Then \mathcal{B}\left(X\right)/\mathcal{K}\left(X\right) is a module. Endow with the metric

\beta \left(T\right)=\beta \left(T{B}_X\right)=d_H\left(T{B}_X,\mathcal{K}\left(X\right)\right)=\underset{K\in \mathcal{K}\left(X\right)}{\mathrm{i}\mathrm{n}\mathrm{f}}{d}_H\left(T{B}_X,K\right),

\mathrm{\forall }T\in B\left(X\right) is derived from d_H on \mathcal{B}\left(X\right)/\mathcal{K}\left(X\right).

The Hausdorff metric can induce a norm \Vert \cdot {\Vert }_H:

\Vert A\Vert =d_H\left(0,A\right)=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\Vert a\Vert :a\in A\right\}.

Under the addition and scalar multiplication defined as above in \mathcal{B}\left(X\right) and \mathcal{B}\left(X\right)/\mathcal{K}\left(X\right) and with the norms induced by the Hausdorff metric, they respectively form normed semigroups.

Let

\mathcal{W}\left(X\right)=\{W\subset X:W\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t}\},

\mathcal{S}\mathcal{W}\left(X\right)=\{S\subset X:S\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{r}\mathrm{a}-\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t}\}.

It can be known from [8, 16] and the Grothendieck theorem that \left(\mathcal{W}\right(X),d_H), and (\mathcal{S}\mathcal{W},d_H) is a closed subspace of \left(\mathcal{B}\left(X\right),d_H\right)[7]. Similarly, a metric can be derived from d_H:

\omega \left(T\right)=d_H(T{B}_X,\mathcal{W}(X))=\underset{W\in \mathcal{W}\left(X\right)}{\mathrm{i}\mathrm{n}\mathrm{f}} d_H(T{B}_X,W),

\tau \left(T\right)=d_H(T{B}_X,\mathcal{S}\mathcal{W}(X))=\underset{S\in \mathcal{S}\mathcal{W}\left(X\right)}{\mathrm{i}\mathrm{n}\mathrm{f}} d_H(T{B}_X,S).

Furthermore, the derived metrics ω(T), τ(T) are respectively called the measure of nonweak compactness of the operator and the measure of non-ultra-weak compactness of the operator on X.

3 Calculations of measure of noncompactness of operators in ℓ^p

Definition 3.1 [14, 15, 24]　Let E₁, E₂ be Banach spaces, and ϕ, ψ be the measures of noncompactness on E₁, E₂, respectively. The map T:D\left(T\right)\subset E_1\to E_2 is continuous.

(a) If there exists k > 0, such that for any nonempty bounded set Q\subset D, there is \psi \left(T\left(Q\right)\right)\le k\varphi \left(Q\right), then T is called a (ϕ, ψ)-contraction map with coefficient k, or simply a k-(ϕ, ψ)-contraction map. When k < 1, it is called a k-(ϕ,ψ)-strict contraction map. In particular, when ϕ = ψ, T is called a k-ϕ-contraction map. Furthermore, when k = 1, T is called a ϕ-contraction map.

(b) If there exists k > 0 such that for any nonempty, bounded, nonrelatively compact set Q\subset D, there is , then T is called a (ϕ, ψ)-condensing map with coefficient k, or simply a k-(ϕ, ψ)-condensing map. When k < 1, it is called a k-(ϕ, ψ)-strict condensing mapping. In particular, when ϕ = ψ, T is called a k-ϕ-condensing map. Furthermore, when k = 1, T is called a ϕ-condensing mapping.

Note 3.1　(a) Clearly, T is a completely continuous map \iffT is a 0-(ϕ, ψ)-contraction map; (ϕ, ψ)-strict contraction map \Rightarrow(ϕ,ψ)-condensing map, but the converse is not true. Examples of condensing maps that are not strict contraction maps can be given [24]. Thus, strict contraction maps and condensing maps are generalizations of completely continuous maps.

(b) When \varphi =\psi =\alpha, T is called a k-set contraction map. When \varphi =\psi =\beta, T is called a k-ball contraction map.

Let E₁, E₂ be Banach spaces, and ϕ, ψ be the measures of noncompactness on E₁, E₂ respectively. The map T:D\left(T\right)\subset E_1\to E_2 is continuous. If T is a k-(ϕ, ψ)-contraction map, \Vert T{\Vert }_{\varphi ,\psi } is defined as follows:

\Vert T{\Vert }_{\varphi ,\psi }=\mathrm{i}\mathrm{n}\mathrm{f}\{k\ge 0:\psi \left(T\left(Q\right)\right)\le k\varphi \left(Q\right),Q\in {\mathcal{M}}_{E_1}\}.

Then \Vert T{\Vert }_{\varphi ,\psi } is called the (ϕ,ψ)-operator norm of T or the (ϕ,ψ)-measure of noncompactness of T. If \varphi =\psi, it is denoted as \Vert T{\Vert }_{\varphi }, which is also equivalent to the following formula:

\Vert T{\Vert }_{\varphi }=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\frac{\varphi \left(T\left(Q\right)\right)}{\varphi \left(Q\right)}:Q\in {\mathcal{M}}_{E_1}\right\}.

Lemma 3.1 [14]　 Let E₁, E₂ be Banach spaces. Then \mathrm{\forall }T\in B\left(E_1,E_2\right),

\Vert T{\Vert }_{\beta }=\beta (T(S_{E_1}))=\beta (T(B_{E_1})).

Lemma 3.2 [19]　 Let E₁, E₂, E₃ be Banach spaces, T\in B\left(E_1,E_2\right),R\in B\left(E_2,E_3\right). Then \Vert T{\Vert }_{\beta } is a seminorm on B\left(E_1,E_2\right), and there are

(1) \Vert T{\Vert }_{\beta }=0\iff T\in \mathcal{K}(E_1,E_2);

(2) \Vert T{\Vert }_{\beta }\le \Vert T\Vert;

(3) \Vert T+K{\Vert }_{\beta }=\Vert T{\Vert }_{\beta },\mathrm{\forall }K\in \mathcal{K}\left(E_1,E_2\right);

(4) \Vert R\circ T\Vert \le \Vert R\Vert \Vert T\Vert.

Lemma 3.3 [14]　 \mathrm{\forall }T\in B(l^1,l^1), (e_i{)}_{i=1}^{\mathrm{\infty }} is the standard basis of l^1. If the representation matrix of T under the basis is A=(a_{ij}{)}_{i,j=1}^{\mathrm{\infty }}, then

\Vert T{\Vert }_{\beta }=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}\sum _{j=n}^{\mathrm{\infty }}\left|a_{jk}\right|.

Corollary 3.1　 \mathrm{\forall }T\in B(l^1,l^1), (e_i{)}_{i=1}^{\mathrm{\infty }} is the standard basis of l^1. If the representation matrix of T under the basis is A=(a_{ij}{)}_{i,j=1}^{\mathrm{\infty }}, then

T\in \mathcal{K}\left(l^1,l^1\right)\iff \Vert T{\Vert }_{\beta }=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}\sum _{j=n}^{\mathrm{\infty }}\left|a_{jk}\right|=0.

Lemma 3.4 [14]　 \mathrm{\forall }T\in B(c_0,c_0), (e_i{)}_{i=1}^{\mathrm{\infty }} is the standard basis of c₀. If the representation matrix of T under the basis is A=(a_{ij}{)}_{i,j=1}^{\mathrm{\infty }}, then

\Vert T{\Vert }_{\beta }=\overline{\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}}\sum _{k=1}^{\mathrm{\infty }}\left|a_{jk}\right|.

Corollary 3.2　 \mathrm{\forall }T\in B(c_0,c_0), (e_i{)}_{i=1}^{\mathrm{\infty }} is the standard basis of c₀. If the representation matrix of T under the basis is A=(a_{ij}{)}_{i,j=1}^{\mathrm{\infty }}, then

T\in \mathcal{K}\left(c_0,c_0\right)\iff \Vert T{\Vert }_{\beta }=\underset{j\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\sum _{k=1}^{\mathrm{\infty }}\left|a_{jk}\right|=0.

Lemma 3.5 [18]　 Let the operator T:C\left[0,1\right]\to C\left[0,1\right] satisfying \left(Tx\right)\left(t\right)=f\left(t\right)x\left(t\right), where f:\left[0,1\right]\to \mathbb{R} is a continuous function. Then

\Vert T{\Vert }_{\beta }=\underset{t\in \left[0,1\right]}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|f\left(t\right)\right|.

Denote ω the set of all sequences of real or complex numbers x=(x_k{)}_{k=1}^{\mathrm{\infty }}. Let e_j be the unit vector with 1 in the j-th position and 0 in all other positions.

Definition 3.2 [2, 22]　A Banach space X\subset \omega is called a BK space if \mathrm{\forall }x=(x_k{)}_{k=1}^{\mathrm{\infty }}\in X, and the projection operator P_n:x\to \mathbb{C},P_{n}x=x_n, P_n will be continuous.

For example, c, c₀, and ℓ^∞ are all BK spaces.

Definition 3.3 [2]　A BK space is said to have the AK property if \mathrm{\forall }x={\left(x_k\right)}_{k=1}^{\mathrm{\infty }}\in X, there is a unique representation x=\sum _{n=1}^{\mathrm{\infty }} x_n{e}_n.

For example, c₀, all possess the AK property.

Let A={\left(a_{nk}\right)}_{n,k=1}^{\mathrm{\infty }} be an infinite dimensional matrix, x\in \omega, andA_n={\left(a_{nk}\right)}_{k=1}^{\mathrm{\infty }} be the nth row of the matrix A, A_{n}x=\sum _{k=1}^{\mathrm{\infty }} a_{nk}{x}_k,Ax=(A_{n}x{)}_{n=1}^{\mathrm{\infty }}. Let X, Y\subset \omega. Denote (X, Y) as the set of all matrices from X to Y. Then A\in \left(X,Y\right) if and only if for all x\in X, n\in \mathbb{N} has A_n\left(x\right) convergence, and A\left(x\right)\in Y.

When X is a Banach space,\beta \left(T\right)=\beta \left(T{B}_X\right)=\beta \left(T{S}_X\right) is called the ball operator measure of noncompactness.

Theorem 3.1　 Let , and the bounded linear map T:{\ell }^p\to {\ell }^p. Then T can be represented by the following infinite matrix (k_ij):

T\left(x\right)\left(i\right)=\sum _{j=1}^{\mathrm{\infty }}{k}_{ij}x\left(j\right), i=1,2,\dots ,\mathrm{\forall }x\in {\ell }^p.

Proof　Let e_j denote the j-th unit vector in ℓ^p, that is e_j=\left(0,\dots ,1,0,\dots \right), where the jth position is 1 and the other positions are 0. Let T\left(e_j\right)\left(i\right)=k_{ij}, \mathrm{\forall }x\in {\ell }^p, and assume x_n=\left(x\left(1\right),x\left(2\right),\dots ,x\left(n\right),0,\dots \right). Then there is

x\left(j\right)-x_n\left(j\right)=\{\begin{array}{ll}0,& j\le n,\\ x\left(j\right),& j>n.\end{array}

\mathrm{\forall }x\in {\ell }^p, and \Vert x-x_n{\Vert }_p^p=\sum _{j=n+1}^{\mathrm{\infty }} \left|x\right(j\left){|}^p\right(n\to \mathrm{\infty }), so x_n\to x in ℓ^p, and T\left(x_n\right)\to T\left(x\right). When n\to \mathrm{\infty }, there is

\begin{array}{rl}{\left|T\left(x_n\right)\left(i\right)-T\left(x\right)\left(i\right)\right|}^p& \le \sum _{j=1}^{\mathrm{\infty }}{\left|T\left(x_n\right)\left(j\right)-T\left(x\right)\left(j\right)\right|}^p\\ & \le \Vert T\left(x_n\right)-T\left(x\right){\Vert }_p^p\to 0.\end{array}

For i = 1, 2, ..., n → ∞, there is T\left(x_n\right)\left(i\right)=\sum _{j=1}^n x\left(j\right)k_{ij}, and

T\left(x_n\right)=\sum _{j=1}^{n}x\left(j\right)T\left(e_j\right), T\left(x_n\right)\left(i\right)=\sum _{j=1}^{n}x\left(j\right)k_{ij}.

So

T\left(x\right)\left(i\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} T\left(x_n\right)\left(i\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \sum _{j=1}^n k_{ij}x\left(j\right)=\sum _{j=1}^{\mathrm{\infty }} k_{ij}x\left(j\right).

□

Theorem 3.2　 Let . Then \mathrm{\forall }T\in B({\ell }^p,{\ell }^p),\mathrm{\exists }A=(a_{ij}{)}_{i,j=1}^{\mathrm{\infty }}\in ({\ell }^p,{\ell }^p), such that

\beta \left(T\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}.

Proof　From Theorem 3.1, \mathrm{\forall }T\in B({\ell }^p,{\ell }^p),\mathrm{\exists }A\in ({\ell }^p,{\ell }^p), satisfying

Tx=Ax, \mathrm{\forall }x\in {\ell }^p.

And

\beta \left(T\right)=\beta \left(T\left(B_X\right)\right)=\beta \left(T\left(S_X\right)\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{x\in S_X}{\mathrm{s}\mathrm{u}\mathrm{p}}\Vert \left(I-P_n\right)\left(Ax\right)\Vert .

Assume P_n:{\ell }^p\to {\ell }^p(n\in \mathbb{N}),P_n\left(y\right)=(y_1,y_2,y_3,\dots ,y_n,0,\dots ),\mathrm{\forall }y\in {\ell }^p, there is

\begin{array}{rl}\Vert \left(I-P_n\right)\left(Ax\right)\Vert & ={\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|A_{m}x\right|}^p\right)}^{\frac{1}{p}}={\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|\sum _{k=0}^{\mathrm{\infty }}{a}_{mk}{x}_k\right|}^p\right)}^{\frac{1}{p}}\\ & \le \sum _{k=0}^{\mathrm{\infty }}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}{x}_k\right|}^p\right)}^{\frac{1}{p}}\le \Vert x\Vert \underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}\\ & =\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}.\end{array}

So

\beta \left(T\right)\le \underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}.

On the other hand, assume e_k represents the kth unit vector of ℓ_p, A{e}_k=(a_{nk}{)}_{n=0}^{\mathrm{\infty }}, and let E=\{e_k:k\in \mathbb{N}\}. Then

\beta \left(AE\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|A_m{e}_k\right|}^p\right)}^{\frac{1}{p}}=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}\le \beta \left(T\right).

Hence,

\beta \left(T\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}.

□

Corollary 3.3　 Let , \mathrm{\forall }T\in B({\ell }^p,{\ell }^p), then T\in \mathcal{K}\left({\ell }^p,{\ell }^p\right) if and only if

\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}=0.

Lemma 3.6 [22]　 Assume X, Y is a BK space, then there are

(a) For every matrix A\in (X,Y), \mathrm{\exists }T\in B(X,Y), such that Tx=Ax,\mathrm{\forall }x\in X;

(b) If X possesses the AK property, \mathrm{\forall }T\in B(X,Y),\mathrm{\exists }A\in (X,Y), there is Tx=Ax, and \mathrm{\forall }x\in X.

Corollary 3.4　 Let X be a BK space with the AK property. Then , such that

\beta \left(T\right)=\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{k}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}.

Corollary 3.5　 Let X be a BK space with the AK property, 1≤p<∞, then T\in \mathcal{K}(X,{\ell }^p) if and only if \mathrm{\exists }A=(a_{ij}{)}_{i,j=1}^{\mathrm{\infty }}\in (X,{\ell }^p), such that

\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left(\sum _{m=n+1}^{\mathrm{\infty }}{\left|a_{mk}\right|}^p\right)}^{\frac{1}{p}}=0.

4 Several equivalent measures of noncompactness of operators

This section presents several common equivalent measures of noncompactness of operators and relevant proofs.

Definition 4.1　Let μ, v be two measures of noncompactness on the Banach space X. If there exist constants a,b>0, \mathrm{\forall }A\subset \mathcal{B}\left(X\right), such that

a\mu \left(A\right)\le \nu \left(A\right)\le b\mu \left(A\right),

then measures of noncompactness μ and ν are equivalent.

Definition 4.2　Let X be a Banach space, B_X be the unit ball, \mathrm{\forall }T\in B\left(X\right), α, β are the set measure of noncompactness and the ball measure of noncompactness in X, respectively. Then \alpha \left(T\right)=\alpha \left(T{B}_X\right),\beta \left(T\right)=\beta \left(T{B}_X\right) are called the measure of noncompactness of the set operator and the measure of noncompactness of the ball operator on X.

According to \beta \left(A\right)\le \alpha \left(A\right)\le 2\beta \left(A\right),\mathrm{\forall }A\in \mathcal{B}\left(X\right), α(T) and β(T) are equivalent, and satisfy \beta \left(T\right)\le \alpha \left(T\right)\le 2\beta \left(T\right).

Theorem 4.1　 Let X={\ell }^{\mathrm{\infty }}, and there is \alpha \left(T\right)=2\beta \left(T\right).

Proof　There is \alpha \left(T\right)\le 2\beta \left(T\right). On the other hand, by the definition, \mathrm{\forall }\epsilon >0,, \mathrm{\exists }{A}_i\in \mathcal{B}\left(X\right), satisfying

T{B}_X\subset \bigcup _{i=1}^n A_i,\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left(T{B}_X\right)\le \alpha \left(T\right)+\epsilon .

For k\in \mathbb{N}, let a_k^i=\mathrm{i}\mathrm{n}\mathrm{f}\{x_k:x=(x^j{)}_{j\in \mathbb{N}}\in A_i\},b_k^i=\mathrm{s}\mathrm{u}\mathrm{p}\{x_k:x=(x^j{)}_{j\in \mathbb{N}}\in A_i\}. If c_k^i=\frac{a_k^i+b_k^i}{2},B_i=B\left(c_k^i,\frac{\alpha \left(T\right)+\epsilon }{2}\right), then A_i\subset B_i, so T{B}_X\subset \bigcup _{i=1}^n B_i, \beta \left(T\right)\le \frac{\alpha \left(T\right)+\epsilon }{2}, and according to the arbitrariness of ε, there is 2\beta \left(T\right)\le \alpha \left(T\right).

□

Theorem 4.2　 Let X be a Banach space, \mathrm{\forall }T\in B\left(X\right), there is \beta \left(T\right)\le \mu \left(T\right), and \mu \left(T^{\ast }\right)\le \mu \left(T\right). Furthermore, if X has a Schauder basis \{e_i{\}}_{i=1}^{\mathrm{\infty }}, P_n is the projection onto \stackrel{-}{span}\{e_i{\}}_{i=n}^{\mathrm{\infty }}, and if {lim}_n\to \mathrm{\infty }\Vert P_n\Vertexists, then \beta \left(T\right) and \mu (T) are equivalent, where \mu \left(T\right)=inf\{\Vert T-K\Vert ,K\in \mathcal{K}\left(X\right)\}.

Proof　Based on y=T{B}_X, there is \mathrm{\exists }x\in B_X, and since y=Tx,\Vert y\Vert \le \Vert Tx\Vert \le \Vert T\Vert, and T{B}_X\subset \Vert T\Vert B_X,\beta \left(T\right)=\beta \left(T{B}_X\right)\le \Vert T\Vert \beta \left(B_X\right)=\Vert T\Vert . Moreover, \beta \left(T\right)=\beta \left(T-K\right)\le \Vert T-K\Vert, so \beta \left(T\right)\le \mu \left(T\right). And

\begin{array}{rl}\mu \left(T^{\ast }\right)& =\mathrm{i}\mathrm{n}\mathrm{f}\{\Vert T^{\ast }+K^{\ast }\Vert ,K^{\ast }\in \mathcal{K}\left(X^{\ast }\right)\}\\ & \le \mathrm{i}\mathrm{n}\mathrm{f}\{\Vert T^{\ast }+K^{\ast }\Vert ,K\in \mathcal{K}\left(X\right)\}\\ & =\mathrm{i}\mathrm{n}\mathrm{f}\{\Vert {\left(T+K\right)}^{\ast }\Vert ,K\in \mathcal{K}\left(X\right)\}\\ & =\mathrm{i}\mathrm{n}\mathrm{f}\{\Vert T+K\Vert ,K\in \mathcal{K}\left(X\right)\}\\ & =\mu \left(T\right).\end{array}

On the other hand,

\begin{array}{rl}\mu \left(T\right)& \le \underset{n}{\mathrm{i}\mathrm{n}\mathrm{f}}\Vert P_{n}T\Vert =\underset{n}{\mathrm{i}\mathrm{n}\mathrm{f}}\underset{x\in S_X}{\mathrm{s}\mathrm{u}\mathrm{p}}\Vert P_{n}Tx\Vert \\ & \le \underset{n}{\mathrm{i}\mathrm{n}\mathrm{f}}\underset{x\in S_X}{\mathrm{s}\mathrm{u}\mathrm{p}}\left(\Vert P_{n}T\left(x\right)-P_n{y}_k\left(x\right)\Vert +\Vert P_n{y}_k\left(x\right)\Vert \right),\end{array}

where y_k\left(x\right)\in \{y_k\left(x\right){\}}_{k=1}^n is a [β(T) + ε]-net of the set TS_X, and it satisfies

\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(y_k,Tx\right)\le \beta \left(T\right)+\epsilon , x\in S_X.

Let t={\underset{\_}{lim}}_{n\to \mathrm{\infty }}\Vert P_n\Vert, and there is

\begin{array}{rl}\mu \left(T\right)& \le \underset{n}{\mathrm{i}\mathrm{n}\mathrm{f}}\left(\Vert P_n\Vert \left(\beta \left(T\right)+\epsilon \right)+\Vert P_n{y}_k\left(x\right)\Vert \right)\\ & \le \underset{n}{\mathrm{i}\mathrm{n}\mathrm{f}}\left[\Vert P_n\Vert \left(\beta \left(T\right)+\epsilon \right)\right]\\ & =t\beta \left(T\right).\end{array}

Then μ(T) and β(T) are equivalent. □

Note 4.1　(a) When t={\underset{\_}{lim}}_{n\to \mathrm{\infty }}\Vert P_n\Vert =1, if or c₀, there is \mu \left(T\right)=\beta \left(T\right).

(b) When X is a reflexive space, and there is \mu \left(T^{\ast }\right)=\mu \left(T\right), and \beta \left(T^{\ast }\right)=\beta \left(T\right).

Theorem 4.3　 Let X be a Banach space, then \mathrm{\forall }T\in B\left(X\right), \beta \left(T^{\ast \ast }\right)\le \beta \left(T\right)\le 2\beta \left(T^{\ast \ast }\right); further, if X is 1-complementable in X**, especially when X is a reflexive space, there is \beta \left(T^{\ast \ast }\right)=\beta \left(T\right).

Proof　Let \beta \left(T\right)=a,\mathrm{\forall }\epsilon >0,(x_i{)}_{i=1}^n is a finite \left(a+\epsilon \right)-net of the set T(B_X). Then there is

T{B}_X\subset \bigcup _{i=1}^n (y_i+(a+\epsilon \left)B_X\right).

Since B_X is w^{\ast }-dense in B_{X^{\ast \ast }} and T^{\ast \ast } is W^{\ast }-continuous, it follows that

\begin{array}{rl}{T}^{\ast \ast }{B}_{X^{\ast \ast }}& =T^{\ast \ast }(\overline{B_X}{)}^{w^{\ast }}={\overline{{T^{\ast \ast }B}_X}}^{w^{\ast }}\\ & ={\overline{{TB}_X}}^{w^{\ast }}\subset {\overline{\bigcup _{i=1}^n\left(yi+\left(a+\epsilon \right)B_X\right)}}^{w^{\ast }}\\ & =\bigcup _{i=1}^n \left(x_i^{\ast }+\left(a+\epsilon \right)B_{X^{\ast \ast }}\right),\end{array}

then \beta \left(T^{\ast \ast }\right)\le \beta \left(T\right).

On the other hand, let \beta \left(T^{\ast \ast }\right)=b, and there exists x_i^{\ast }\in X^{\ast \ast }, such that

T^{\ast \ast }{B}_{X^{\ast \ast }}\subset \bigcup _{i=1}^n B(x_i^{\ast }+b),

and T{B}_X\subset T^{\ast \ast }{\overline{B_X}}^{w^{\ast }}, so T{B}_X\subset \bigcup _{i=1}^n B(x_i^{\ast },b), and there is

T{B}_X\cap X\subset \left(\bigcup _{i=1}^n B(x_i^{\ast },b)\right)\cap X=\bigcup _{i=1}^n \left(B\right(x_i^{\ast },b)\cap X).

LetA_i=B\left(x_i^{\ast },b\right)\cap X,\mathrm{d}\mathrm{i}\mathrm{m}{A}_i\le 2b, and there exists x_i\in X such that \left|x_i-x_i^{\ast }\right|\le 2b. \bigcup _{i=1}^n A_i\subset \bigcup _{i=1}^n B(x_i^{\ast },2b), so, T{B}_X\subset \bigcup _{i=1}^n B(x_i^{\ast },2b). \beta \left(T\right)\le 2b=2\beta \left(T^{\ast \ast }\right).

When X is 1-supplementable in X**, let

P_i:X^{\ast \ast }\to X,

P_i\left(x_i^{\ast \ast }\right)\mapsto x_i.

For u_i\in A_i,x_i\in X, there is

\left|u_i-x_i\right|=\left|P_i{u}_i-P_i{x}_i^{\ast }\right|\le \Vert P_i\Vert \left|u_i-x_i^{\ast }\right|\le b,

and {\cup }_{i=1}^n{A}_i\subset {\bigcup }_{i=1}^{n}B\left(x_i,b\right), so T{B}_X\subset {\cup }_{i=1}^{n}B\left(x_i,b\right), and

\beta \left(T\right)\le b=\beta \left(T^{\ast \ast }\right).

Corollary 4.1　 Let X be a Banach space, then \mathrm{\forall }T\in B\left(X,X\right), and \beta \left(T^{\ast }\right)=\beta \left(T^{\ast \ast \ast }\right).

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