Point localization and spectral theory for symmetric random operators

In a Hilbert space context, we propose a rather general notion of “random operators” which allows for taking stochastic limits. After establishing a connection with measurable fields of closed operators, we may speak of a spectral theory for symmetric random operators. Received: 18 December 2008     

Xia Spectrum for Some Class of Operators

In this paper, we introduce Xia spectra of n-tuples of operators satisfying |T ²| ≥ U|T ²|U* for the polar decomposition of T = U|T| and we extend Putnam’s inequality to these tuples [7]. This research is partially supported by Grant-in-Aid Research No.17540176.     

Localizable spectrum and bounded local resolvent functions

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown that, for any measure with natural spectrum on a locally compact abelian group, the corresponding operator of convolution on the group algebra admits a non-zero bounded local resolvent function precisely when its spectrum has non-empty interior. Received: 15 November 2007     

Spectral inclusions for operators on Banach spaces

This article centers around the relation between the spectra of two Banach space operators that are linked by some intertwining condition such as quasi-similarity. Certain conditions from local spectral theory are shown to be both necessary and sufficient for these operators to have equal spectra, approximate point spectra, or surjectivity spectra. A key role is played by a localized version of Bishop’s classical property (β) and a related closed range condition. As an application to harmonic analysis, the measures on a locally compact abelian group that avoid the Wiener-Pitt phenomenon are characterized in terms of local spectral theory.     

Algebraic extensions of semi-hyponormal operators

In this paper, we show that algebraic extensions of semi-hyponormal operators (defined below) are subscalar. As corollaries we get the following:

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

On The Extended Eigenvalues of Some Volterra Operators

We show that a large class of compact quasinilpotent operators has extended eigenvalues. As a consequence, if V is such an operator, then the associated spectral algebra contains its commutant {V}' as a proper subalgebra.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Essential Spectral Inclusions for Operators on Banach Spaces

This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context.      

Polaroid operators,SVEP and perturbed Browder,Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T ^*+A ^*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).     

Generalized Browder’s Theorem and SVEP

A bounded operator a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H ₀(λI − T) as λ belongs to certain subsets of . In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators.     

A Remark on Support of the Principal Function for Class A Operators

Let be an invertible class A operator such that . Then we show that , where gT is the principal function of T. Moreover, we show that if T is pure, then .     

The boundedness below of 2×2 upper triangular operator matrices

Wen and are given we denote byM _C an operator acting on the Hilbert space of the form

Perturbation theorems for holomorphic semigroups

The concept of the gap function is used to give new perturbation results for generators of holomorphic semigroups. In particular, we show that if A is the generator of a holomorphic semigroup on a Banach space and , then every closed linear operator C such that for some and

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

Some Properties of Essential Spectra of a Positive Operator

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ_ew(T) of the operator T is a set , where is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum

The fundamental theorem of geometric calculus via a generalized riemann integral

Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative ∇ _V F exists and is Lebesgue integrable. We also give sufficient conditions that ∇ _V F exists.     

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

Additive Maps Preserving Local Spectrum

Let X be a complex Banach space, and let be the space of bounded operators on X. Given and x ∈ X, denote by σ_T (x) the local spectrum of T at x. We prove that if is an additive map such that

Linearization,trace, and determinant of certain meromorphic operator functions

In this paper operator functions of type