A Functional Calculus for Pairs of Commuting Contractions (original) (raw)

Measures on the unit circle and unitary truncations of unitary operators

Journal of Approximation Theory, 2006

In this paper we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator.

Spectral Decomposition Of A Functional Calculus For Noncommuting Systems Of Operators

For a nite system A of bounded linear operators whose real linear combinations have real spectrum, the monogenic functional calculus is used to associate a function f(A) of the operators with any analytic function f in n real variables de ned in a neighbourhood of the monogenic spectrum (A). Here, the existence of projection operators that decompose the functional calculus onto invariant subspaces associated with each connected component of the monogenic spectrum (A) is shown by employing Cauchy's integral formula in Cli ord analysis.

A "typical" contraction is unitary

L’Enseignement Mathématique, 2010

We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator topology as well. These results are applied to the problem of embedding operators into strongly continuous semigroups.

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)

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...

Spectral measures in -algebras of singular integral operators with shifts

Journal of Functional Analysis, 2007

The C *-subalgebra B of B(L 2 (T)) generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms which acts topologically freely on T is studied. A symbol calculus for the C *-algebra B and a Fredholm criterion for the operators B ∈ B are established by using a generalization of the local-trajectory method for studying C *-algebras associated with C *-dynamical systems. This method is related to the Allan-Douglas local principle and its generalization is based on the notion of spectral measure.

2 Operator-Valued Measures, Dilations, and the Theory of Frames

2016

We develop elements of a general dilation theory for operatorvalued measures. Hilbert space operator-valued measures are closely related to bounded linear maps on abelian von Neumann algebras, and some of our results include new dilation results for bounded linear maps that are not necessarily completely bounded, and from domain algebras that are not necessarily abelian. In the non-cb case the dilation space often needs to be a Banach space. We give applications to both the discrete and the continuous frame theory. There are natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of continuous linear maps between C *-algebras. In this connection frame theory itself is identified with the special case in which the domain algebra for the maps is an abelian von Neumann algebra and the map is normal (i.e. ultraweakly, or σ-weakly, or w*) continuous. Some of the results for maps extend to the case where the domain algebra is non-commutative. It has been known for a long time that a necessary and sufficient condition for a bounded linear map from a unital C*-algebra into B(H) to have a Hilbert space dilation to a *-homomorphism is that the mapping needs to be completely bounded. Our theory shows that even if it is not completely bounded it still has a Banach space dilation to a homomorphism. For the special case when the domain algebra is an abelian von Neumann algebra and the map is normal, we show that the dilation can be taken to be normal with respect to the usual Banach space version of ultraweak topology on the range space. We view these results as generalizations of the known result of Cazzaza, Han and Larson that arbitrary framings have Banach dilations, and also the known result that completely bounded maps have Hilbertian dilations. Our methods extend to some cases where the domain algebra need not be commutative, leading to new dilation results for maps of general von Neumann algebras. This paper was motivated by some recent results in frame theory and the observation that there is a close connection between the analysis of dual pairs of frames (both the discrete and the continuous theory) and the theory of operator-valued measures.