A Criterion for an Operator with Real Spectrum to Be of Scalar-type.
Compactness properties of the integration mapassociated with a vector measure
Comparison of joint spectra for certain classes of commuting operators
Spectral representation of local semigroups in locally convex spaces
Sequential closures of -subalgebras for a vector measure
Let be a locally convex space, be a vector measure defined on a -algebra , and be the associated (locally convex) space of -integrable functions. Let denote , equipped with the relative topology from . For a subalgebra , let denote the generated -algebra and denote the closure of in . Sets of the form arise in criteria determining separability of ; see [6]. We consider some natural questions concerning and, in particular, its relation to . It is shown that and moreover,...
Bade's theorem on the uniformly closed algebra generated by a Boolean algebra
Examples of stable -semigroups
The Uniformly Closed Algebra Generated by a Complete Boolean Algebra of Projections.
Optimal domains for the kernel operator associated with Sobolev's inequality
Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0,1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain....
Corrigenda to: "Optimal domains for the kernel operator associated with Sobolev's inequality" (Studia Math. 158 (2003), 131-152)
Criteria for weak compactness of vector-valued integration maps
Criteria are given for determining the weak compactness, or otherwise, of the integration map associated with a vector measure. For instance, the space of integrable functions of a weakly compact integration map is necessarily normable for the mean convergence topology. Results are presented which relate weak compactness of the integration map with the property of being a bicontinuous isomorphism onto its range. Finally, a detailed description is given of the compactness properties for the integration...
Operator ideal properties of vector measures with finite variation
Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the...
Characterizing Fréchet-Schwartz spaces via power bounded operators
We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet-Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a Fréchet-Schwartz space does so in a special way. We single out this type of "rapid convergence" for a sequence...
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