Hausdorff measures of noncompactness and interpolation spaces

2015

A new measure of weak noncompactness is introduced. A log-arithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of operators T: Ai → Bi, i = 0, 1 is weakly compact, then so is T: Aθ,p → Bθ,p for all 0 < θ < 1 and 1 < p <∞.

Acta Mathematica Hungarica, 2010

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.

Nonlinear Analysis-theory Methods & Applications, 2010

In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary BK -space into the sequence spaces c 0 , c, ∞ and 1 , and into the matrix domains of triangles in these spaces. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on the BK -spaces.

Mathematica Slovaca, 1992

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Journal of Mathematical Analysis and Applications, 2020

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Transactions of the American Mathematical Society, 2018

Working in the setting of quasi-Banach couples, we establish a formula for the measure of noncompactness of bilinear operators interpolated by the general real method. The result applies to the real method and to the real method with a function parameter. c

Journal of Mathematical Analysis and Applications, 2006

In this paper, we characterize some operators and matrix transformations in the sequence spaces s α , s 0 α , s (c) α , p α. Moreover, using the Hausdorff measure of noncompactness necessary and sufficient conditions are formulated for a linear operator between the mentioned spaces to be compact. Among other things, some results of Cohen and Dunford are recovered.

Journal of Convex Analysis, 2014

Let E be a non-Archimedean Banach space over a non-Archimedean locally compact nontrivially valued field K := (K, |.|). Let E ′′ be its bidual and M a bounded set in E. We say that M is ε−weakly relatively compact if M σ(E ′′ ,E ′) ⊂ E+B E ′′ ,ε , where B E ′′ ,ε is the closed ball in E ′′ with the radius ε ≥ 0. In this paper we describe measures of noncompactness γ, k and De Blasi measure ω. We show that γ (M) ≤ k (M) ≤ ω (M) = ω(aco M) ≤ 1 |ρ| γ (M), where ρ (|ρ| < 1) is an uniformizing element in K, and ω(M) = sup{lim m dist (x m , [x 1 ,. .. , x m−1 ]) : (x m) ⊂ M }; the latter equality is purely non-Archimedean. In particular, assuming |K| = {||x|| : x ∈ E}, we prove that the absolutely convex hull aco M of a ε−weakly relatively compact subset M in E is ε−weakly relatively compact. In fact we show that in this case for a bounded set M in E we have γ (M) = γ (aco M) = k (M) = k(aco M) = ω (M). Note that the above equalities fail in general for real Banach spaces by results of Granero ([9]) and Astalla and Tilly ([5]). Most of proofs are strictly non-Archimedean. A non-Archimedean variant of another quantitative Krein's theorem due to Fabian, Hajek, Montesinos and Zizler is also provided, see Corollary 3.9.

Abstract. In this paper, we derive some identities for the Hausdorff measures of noncompactness of certain matrix operators on the sequence spaces X (r, s) of generalized means. Further, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for such operators to be compact.

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.