Spectral Measures In Abstract Spaces (original) (raw)

Spectral measures, II: Characterization of scalar operators

Periodica Mathematica Hungarica, 1982

We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be "spectral". A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A R a d o n-N i k o d y m theorem is proved.

Banach function spaces and spectral measures

1982

The fundamental link between prespectral measures and Banach function spaces is to be found in a theorem of T.A.Gillespie which relates cyclic spaces isornorphically to certain Banach function spaces. We obtain here an extension of this result to the wider class of precyclic spaces. We then consider the properties of weak sequential completeness and reflexivity in Banach function spaces: necessary and sufficient conditions are obtained which in turn, via the afore-mentioned isomorphisms., both extend and simplify analogously formulated existing results for cyclic spaces. Finally the concept of a homomorphism between pairs of Banach function spaces is examined.The class of such mappings is determined and a complete description obtained in the form of a (unique) disjoint sum of two mappings, one of which is always an isomorphism and the other of which is arbitrary in a certain sense, or null.It is shown moreover that the isomorphic component itself is composed of two other isomorphism...

Spectral measures over c-algebras of operators defined in c_0c_0c_0

arXiv: Functional Analysis, 2014

The main goal of this work is to introduce an analogous in the non-archimedean context of the Gelfand spaces of certain Banach commutative algebras with unit. In order to do that, we study the spectrum of this algebras and we show that, under special conditions, these algebras are isometrically isomorphic to certain spaces of continuous functions defined over compacts. Such isometries preserve projections and allow to define associated measures which are known as spectral measures. We also show that each element of the algebras can be represented as an integral defined by these measures. We finish this work by showing that the studied algebras are, actually, free Banach spaces. 1. Introduction and notation Many researchers have tried to generalize the elemental studies of Banach alge- bras from classical case to vectorial structures over non-archimedean fields. The first big task was to find a results similar to the Gelfand-Mazur Theorem in this context. But, this theorem failed sin...

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.

Some results about spectral theory over Fréchet spaces

2021

In this study, we present some differences that arise in the spectral analysis of pseudodifferential operators with constant coefficients, when we use the Fr\'echet structure instead of the Banach structure. Here we show that this change in the topology implies in significant changes on the behavior of the Laplace's spectrum, for instance, the resolvent set vanishes, even with a bounded domain. The notion of exponential dichotomy for Fr\'echet Spaces introduced by the author and its connection with the separation of spectrum and dichotomy inspired us to make this work.

Measures on topological spaces

Journal of Mathematical Sciences, 1998

Integration on topological spaces is a field of mathematics which could be defined as the intersection of functional analysis, general topology, and probability theory. However, at different epochs the roles of these three ingredients were different, and, moreover, very often none of the three exerted a dominating'influence. For example, the theory of topological groups and analysis on manifolds gave rise to questions concerning Haar measures, Riemannian volumes, and other measures on locally compact spaces, and their influence was so strong that until recently many fundamental books on integration dealt exclusively with locally compact spaces. On the other hand, quantum fields and statistical physics provide problems of a totally different type, and this circumstance results in another trend in the theory of integration. At present, measure theory is especially strongly influenced by the intensive developmdnt of infinite-dimensional analysis in a broad sense, including stochastic analysis, dynamic systems, and the theory of representations of groups. This development involves measures on complicated infinite-dimensional manifolds and functional spaces. Recent investigations in population genetics have given rise to measure-valued diffusions, which, in turn, lead to such objects as measures on spaces of measures.

Duality Questions for Operators, Spectrum and Measures

Acta Applicandae Mathematicae, 2009

We explore spectral duality in the context of measures in R n , starting with partial differential operators and Fuglede's question (1974) about the relationship between orthogonal bases of complex exponentials in L 2 (Ω) and tiling properties of Ω, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in R n , formulated first by Jorgensen and Pedersen.

Review: Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions

Bulletin (New Series) of the American Mathematical …, 1986

A linear transformation T acting on a finite-dimensional complex vector space SC can always be decomposed as T = D + N, where (i) D is diagonalizable and N is nilpotent; and (ii) DN = ND\ moreover, such a decomposition is unique with respect to the conditions (i) and (ii), and both D and N are indeed polynomials in T. When % is an infinite-dimensional Banach space, such a representation for a bounded operator T is no longer true, but an important class of transformations introduced and studied by N. Dunford [3] in the 1950s possesses a similar property. By definition, a spectral operator T acting on 3C is one for which there exists a spectral measure E (i.e., a homomorphism from the Boolean algebra of Borel subsets of the complex plane C into the Boolean algebra of projection operators on & such that E is bounded and E(C) = /) satisfying the following two properties: (1) TE(B) = E(B)T\ and (2) a(T\ E(B)sr)