Weak compactness in the space of operator valued measures Mba(S, L(X,Y)) and its applications (original) (raw)

Measures of weak non-compactness in spaces of nuclear operators

Mathematische Zeitschrift, 2019

We show that in the space of nuclear operators from ℓ q (Λ) to ℓ p (J) the two natural ways of measuring weak non-compactness coincide. We also provide explicit formulas for these measures. As a consequence the same is proved for preduals of atomic von Neumann algebras.

A Note on Radon-Nikodym Theorem for Operator Valued Measures and Its Applications

Communications of the Korean Mathematical Society

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

Strongly Bounded Representing Measures and Convergence Theorems

Glasgow Mathematical Journal, 2010

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Vector measures and the strong operator topology

Proceedings of The American Mathematical Society, 2009

A fundamental result of Nigel Kalton is used to establish a result for operator valued measures which has improved versions of the Vitali-Hahn-Saks Theorem, Phillips's Lemma, the Orlicz-Pettis Theorem and other classical results as straightforward corollaries.

Weakly compact operators and multipliers on C0(S, A)

Journal of Mathematical Analysis and Applications, 1983

The problem of defining a convolution of operator-valued measures defined on a locally compact semitopological semigroup which represent weakly compact operators is considered in this paper. Convolution of the corresponding weakly compact operators is then defined. A characterization of multipliers on function spaces is given in terms of their representing measures.

On weak compactness in L 1 spaces

We will use the concept of strong generating and a simple renorm-ing theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L 1 (µ) spaces for finite measures µ.

Weakly compact operators and u-additive measures

Annales mathématiques Blaise Pascal, 2000

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Weak convergence in spaces of measures and operators

Bulletin of the Belgian Mathematical Society - Simon Stevin, 1999

J. K. Brooks and P. W. Lewis have established that if E and E * have RNP, then in M (Σ, E), m n converges weakly to m if and only if m n (A) converges weakly to m(A) for each A ∈ Σ. Assuming the existence of a special kind of lifting, N. Randrianantoanina and E. Saab have shown an analogous result if E is a dual space. Here we show that for the space M (P(N), E) where E * is a Grothendieck space or E is a Mazur space, this kind of weak convergence is valid. Also some applications for subspaces of L(E, F) similar to the results of N. Kalton and W. Ruess are given.