Compactness Criteria and Measures of Noncompactness in Function Spaces

TY - JOUR

T1 - Compactness Criteria and Measures of Noncompactness in Function Spaces

AU - Eveson, Simon

PY - 2000/8/1

Y1 - 2000/8/1

N2 - In [9], R. S. Phillips gave a compactness criterion for subsets of a Banach space X, namely, that if (Ts) is a net of compact operators converging strongly to the identity then a bounded set A ¿ X is relatively compact if and only if (Ts) converges uniformly on A to the identity. He then used this general result to give concrete compactness criteria in some specific spaces and to investigate compactness of operators. In this paper, we develop these ideas in two directions: firstly, to demonstrate that this method can be used as a unifying tool to derive many classical compactness criteria originally proved using other techniques, and secondly to extend the method from giving a criterion for compactness to giving a generalised measure of noncompactness. The conditions for compactness that result from this approach are not new (though they are possibly given in slightly more generality than can easily be found in the literature), but they all arise from a single, simple, and usually elementary, approach. Throughout, vector operations applied to sets are all defined pointwise, so if A and B are subsets of a vector space, x is a vector and ¿ is a scalar, then A+B={a+b:a¿A, b¿B}, ¿A={¿a:a¿A}, x+A={x+a:a¿A}. The ball centred at x with radius r in a normed space X is denoted BX(x; r). More generally, if A is a subset of a normed space X then the r-neighbourhood BX(A; r) of A is defined by Bx(A;r)={x¿X: for some a¿A,¿x-a¿<r}. It is sometimes convenient to combine these notations, for example to use x+rBX(0; 1) in place of BX(x; r) or A+rBX(0; 1) in place of BX(A; r). The ball measure of noncompactness in a normed space X will be denoted by ßX: if A is a bounded subset of X then ßX(A) is the infimum of r such that A can be covered with finitely many balls of radius r.

AB - In [9], R. S. Phillips gave a compactness criterion for subsets of a Banach space X, namely, that if (Ts) is a net of compact operators converging strongly to the identity then a bounded set A ¿ X is relatively compact if and only if (Ts) converges uniformly on A to the identity. He then used this general result to give concrete compactness criteria in some specific spaces and to investigate compactness of operators. In this paper, we develop these ideas in two directions: firstly, to demonstrate that this method can be used as a unifying tool to derive many classical compactness criteria originally proved using other techniques, and secondly to extend the method from giving a criterion for compactness to giving a generalised measure of noncompactness. The conditions for compactness that result from this approach are not new (though they are possibly given in slightly more generality than can easily be found in the literature), but they all arise from a single, simple, and usually elementary, approach. Throughout, vector operations applied to sets are all defined pointwise, so if A and B are subsets of a vector space, x is a vector and ¿ is a scalar, then A+B={a+b:a¿A, b¿B}, ¿A={¿a:a¿A}, x+A={x+a:a¿A}. The ball centred at x with radius r in a normed space X is denoted BX(x; r). More generally, if A is a subset of a normed space X then the r-neighbourhood BX(A; r) of A is defined by Bx(A;r)={x¿X: for some a¿A,¿x-a¿<r}. It is sometimes convenient to combine these notations, for example to use x+rBX(0; 1) in place of BX(x; r) or A+rBX(0; 1) in place of BX(A; r). The ball measure of noncompactness in a normed space X will be denoted by ßX: if A is a bounded subset of X then ßX(A) is the infimum of r such that A can be covered with finitely many balls of radius r.

KW - Analysis

U2 - 10.1112/S0024610700008991

DO - 10.1112/S0024610700008991

M3 - Article

VL - 62

SP - 263

EP - 277

JO - J. Lon. Math. Soc

JF - J. Lon. Math. Soc

IS - 1

ER -