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A Borel extension approach to weakly compact operators on C 0 ( T )

Thiruvaiyaru V. Panchapagesan (2002)

Czechoslovak Mathematical Journal

Let X be a quasicomplete locally convex Hausdorff space. Let T be a locally compact Hausdorff space and let C 0 ( T ) = { f T I , f is continuous and vanishes at infinity } be endowed with the supremum norm. Starting with the Borel extension theorem for X -valued σ -additive Baire measures on T , an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map u C 0 ( T ) X to be weakly compact.

A generalized Pettis measurability criterion and integration of vector functions

I. Dobrakov, T. V. Panchapagesan (2004)

Studia Mathematica

For Banach-space-valued functions, the concepts of 𝒫-measurability, λ-measurability and m-measurability are defined, where 𝒫 is a δ-ring of subsets of a nonvoid set T, λ is a σ-subadditive submeasure on σ(𝒫) and m is an operator-valued measure on 𝒫. Various characterizations are given for 𝒫-measurable (resp. λ-measurable, m-measurable) vector functions on T. Using them and other auxiliary results proved here, the basic theorems of [6] are rigorously established.

A note on the construction of measures taking their values in a Banach space with basis.

María Congost Iglesias (1983)

Stochastica

If E is a Banach space with a basis {en}, n belonging to N, a vector measure m: a --> E determines a sequence {mn}, n belonging to N, of scalar measures on a named its components. We obtain necessary and sufficient conditions to ensure that when given a sequence of scalar measures it is possible to construct a vector valued measure whose components were those given. Furthermore we study some relations between the variation of the measure m and the variation of its components.

A scalar Volterra derivative for the PoU-integral

V. Marraffa (2005)

Mathematica Bohemica

A weak form of the Henstock Lemma for the P o U -integrable functions is given. This allows to prove the existence of a scalar Volterra derivative for the P o U -integral. Also the P o U -integrable functions are characterized by means of Pettis integrability and a condition involving finite pseudopartitions.

A simple proof of the Borel extension theorem and weak compactness of operators

Ivan Dobrakov, Thiruvaiyaru V. Panchapagesan (2002)

Czechoslovak Mathematical Journal

Let T be a locally compact Hausdorff space and let C 0 ( T ) be the Banach space of all complex valued continuous functions vanishing at infinity in T , provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X -valued σ -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u C 0 ( T ) X when...

Abstract Perron-Stieltjes integral

Štefan Schwabik (1996)

Mathematica Bohemica

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [4]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In [3] Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig’s integral the Dushnik interior integral...

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