Strongly definitizable linear pencils in Hilbert space

Selfadjoint linear pencils F–G are considered which have discrete spectrum and neither F nor G is definite. Several characterizations are given of a strongly definitizable property when F and G are bounded, and also when both operators are unbounded. The theory is applied to analysis of the stability of a linear second order initial-boundary value problem with boundary conditions dependent on the eigenvalue parameter.Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

On the filling in holes problem for operator matrices

We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces.     

On the continuity of the spectrum in certain Banach algebras

In this paper we describe some classes of linear operatorsTL(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spacesH such that the spectrum (T, L(H)) is continuous at the pointsT from these classes. We also describe some subalgebras of the algebras for which the spectrum (x,) becomes continuous at the pointsx when (x,) is restricted to the subalgebra . In particular, we show that the spectrum (x,) is continuous in Banach algebras with polynomial identities. Examples of such algebras are given.This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Convolution equations on finite intervals and factorization of matrix functions

Systems of convolution equations on a finite interval are reduced to the problem of canonical factorization of unimodular matrix-valued functions. The discrete version is considered separately.This research was partially supported by the Israel Science Foundation funded by the Israel Academy of Sciences and Humanities.     

Browder’s Theorems through Localized SVEP

A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds.     

On closures of joint similarity orbits

For an n-tuple T=(T₁,..., T_n) of operators on a Hilbert spacexxHx, the joint similarity orbit of T isxxSx(T)={VTV^–1 =(VT₁V^–1,...,VT_nV^–1): V is invertible onxxHx}. We study the structure of the norm closure ofxxSx, both in the case when T is commutative and when it is not. We first develop a Rota-model for the Taylor spectrum and use it to study n-tuples with totally disconnected Taylor spectrum, in particular quasinilpotent ones. We then consider limits of nilpotent n-tuples, and of normal n-tuples. For noncommuting n-tuples, we present a number of surprising facts relating the closure ofxxSx(T) to the Harte spectrum of T and the lack of commutativity of T. We show that a continuous function which is constant onxxSx(T) for all T must be constant. We conclude the paper with a detailed study of closed similarity orbits.Research partially supported by grants from the National Science Foundation.     

On Bounded Local Resolvents

It is known that each normal operator on a Hilbert space with nonempty interior of the spectrum admits vectors with bounded local resolvent. We generalize this result for Banach space operators with the decomposition property (δ) (in particular for decomposable operators). Moreover, the same result holds for operators with interior points in the localizable spectrum.     

Filling in the holes of the semi-Fredholm domain

We study the following problem: given a set of holes in the semi-Fredholm domain of an operator, is there an invariant subspace of the operator such that the spectrum of the restriction is equal to the spectrum of the operator together with the set of holes?     

On the Fredholm and Weyl Spectra of Several Commuting Operators

In the paper the local structure of the Fredholm joint spectrum of commuting n-tuples of operators is considered. A connection between the spatial characteristics of operators and the algebraic invariants of the corresponding coherent sheaves is investigated. A new notion of Weyl joint spectrum of commuting n-tuple is introduced.     

Invariant subspaces and localizable spectrum

LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum _loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity.     

On some properties of factorization indices

Various kinds of factorization indices are considered (right partial indices, left partial indices, Birkhoff indices), and some connections between them are described. We solve also the problem on the relation between the partial indices of two matrix functions and of their product.Dedicated to the memory of Mark Grigorievich KreinThis research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

ISOLATED POINTS OF THE APPROXIMATE POINT SPECTRUM OF CERTAIN LATTICE HOMOMORPHISMS ON Co (X)

Abstract

Suppose X is a locally compact Hausdorff space and C (X) the apace of all continuous complex valued functions on X which vanish at infinity. Let T be a (complex) linear lattice homomorphism on C_o (X) whose adjoint is also a lattice homomorphism. It is sham that every non-zero isolated point of the approximate point spectrum of T lies in the point spectrum of T. An example is given to show that the exclusion of zero is necessary, even when X is compact. The same techniques are then used to show that if also the spectrum of T is finite then T can be written, in a natural manner, as a direct sum of two such lattice homomorphisms; one being an n'th root of an invertible multiplication operator and the other quasi-nilpotent.     

Topological uniform descent and Weyl type theorem

The generalized Weyl’s theorem holds for a Banach space operator T if and only if T or T^∗ has the single valued extension property in the complement of the Weyl spectrum (or B-Weyl spectrum) and T has topological uniform descent at all λ which are isolated eigenvalues of T. Also, we show that the generalized Weyl’s theorem holds for analytically paranormal operators.     

Regular Functions of Operators on Tensor Products of Hilbert Spaces

A class of linear operators on tensor products of Hilbert spaces is considered. Estimates for the norm of operator-valued functions regular on the spectrum are derived. These results are new even in the finite-dimensional case. By virtue of the obtained estimates, we derive stability conditions for semilinear differential equations. Applications of the mentioned results to integro-differential equations are also discussed.     

Taylor spectrum of commuting n-contractions

Abstract

In this paper, we characterize the Taylor spectrum for a certain class of commuting n-contractions. We also investigate the behavior of this spectrum under action of involutive automorphisms of the unit ball 𝔹 _n.     

Additive local spectrum compressors

Let L(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize additive continuous maps from L(X) onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector. Additive continuous maps from L(X) onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized.     

Weyl's theorem for operator matrices

Weyl's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison Browder's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl's theorem and Browder's theorem are liable to fail for 2×2 operator matrices. In this paper we explore how Weyl's theorem and Browder's theorem survive for 2×2 operator matrices on the Hilbert space.Supported in part by BSRI-97-1420 and KOSEF 94-0701-02-01-3.     

The spectrum of the continuous Laplacian on a graph

We study the spectrum of the continuous Laplacian on a countable connected locally finite graph without self-loops, whose edges have suitable positive conductances and are identified with copies of segments [0, 1], with the condition that the sum of the weighted normal exterior derivatives is 0 at every node (Kirchhoff-type condition). In particular, we analyse the relation-between the spectrum of the operator and the spectrum of the discrete Laplacian (I-P) defined on the vertices of .     

q-deformed Circular Operators

In this paper, q-deformed circularity of an operator in a Hilbert space is introduced and investigated. In particular, the q-deformed circularity of a weighted shift is characterized by an integer, which is uniquely determined by the operator. It is shown that that the spectrum of a q-deformed circular weighted shift is the whole complex plane.