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Measure of non-compactness of operators interpolated by the real method

Radosław Szwedek (2006)

Studia Mathematica

We study the measure of non-compactness of operators between abstract real interpolation spaces. We prove an estimate of this measure, depending on the fundamental function of the space. An application to the spectral theory of linear operators is presented.

Measure of weak noncompactness under complex interpolation

Andrzej Kryczka, Stanisław Prus (2001)

Studia Mathematica

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is T : A [ θ ] → B [ θ ] for all 0 < θ < 1, where A [ θ ] and B [ θ ] are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.

Measures of noncompactness and normal structure in Banach spaces

J. García-Falset, A. Jiménez-Melado, E. Lloréns-Fuster (1994)

Studia Mathematica

Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.

Measures of non-compactness in Orlicz modular spaces.

A. G. Aksoy, J.-B. Baillon (1993)

Collectanea Mathematica

In this paper we show that the ball-measure of non-compactness of a norm bounded subset of an Orlicz modular space L-Psi is equal to the limit of its n-widths. We also obtain several inequalities between the measures of non-compactness and the limit of the n-widths for modular bounded subsets of L-Psi which do not have Delta-2-condition. Minimum conditions on Psi to have such results are specified and an example of such a function Psi is provided.

Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces

Claudio H. Morales (1992)

Commentationes Mathematicae Universitatis Carolinae

Let X be a real Banach space. A multivalued operator T from K into 2 X is said to be pseudo-contractive if for every x , y in K , u ∈ T ( x ) , v ∈ T ( y ) and all r > 0 , ∥ x - y ∥ ≤ ∥ ( 1 + r ) ( x - y ) - r ( u - v ) ∥ . Denote by G ( z , w ) the set { u ∈ K : ∥ u - w ∥ ≤ ∥ u - z ∥ } . Suppose every bounded closed and convex subset of X has the fixed point property with respect to nonexpansive selfmappings. Now if T is a Lipschitzian and pseudo-contractive mapping from K into the family of closed and bounded subsets of K so that the set G ( z , w ) is bounded for some z ∈ K and some w ∈ T ( z ) , then T has a fixed point in K .

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