Weyl's theorem and quasi-similarity

In this paper we give necessary and sufficient conditions of a finitely ascensive operator to obey Weyl's theorem and study Weyl's theorem for quasi-affine transforms. In particular, we give an improvement of the earlier result of Duggal forp-hyponormal operators.     

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.     

The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices

When A∈B(H) and B∈B(K) are given, we denote by M_C the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form In this paper, for given A and B, the sets and ?_C∈Inv(K,H)σ_l(M_C) are determined, where σ_l(T),B_l(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H.     

On Fredholm index in Banach spaces

We prove the stability of the index of a Fredholm complex of Banach spaces under those compact perturbations which are uniform limits of finite-rank operators (Theorem 3.3). This result is a consequence of some similar statements (Theorems 3.1 and 3.2) concerning more general objects, namely the Fredholm pairs (Definition 1.1).     

Dunford's property (C) and Weyl's theorems

This paper studies Weyl's theorems, and some related results for operators with Dunford's property (C). Weyl's theorem in some classes of operators (e.g.M-hyponormal,p-hyponormal and totally paranormal operators) is considered.     

Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Essential approximate point spectra and Weyl's theorem for operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H⊕K of the form . In this paper, it is shown that a 2×2 operator matrix MC is upper semi-Fredholm and ind(MC)?0 for some C∈B(K,H) if and only if A is upper semi-Fredholm and

     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

On the Friedrichs extension of some block operator matrices

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2×2 operator matrices and study their essential spectrum.     

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.     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.     

Completion of operator partial matrices associated with chordal graphs

H. Dym and I. Gohberg established in [6] necessary and sufficient conditions for the existence and uniqueness of an invertible block matrix F=(F_ij)_i,j=1,...,n such that F_ij=R_ij for |i–j|m and F^–1 has a band triangular factorization and so (F^–1)_ij=0 for |i–j|>m. Here R_ij, |i–j|m are given block matrices.The aim of this paper is to generalize these results in two directions. First, we shall allow R_ij to be an (linear bounded) operator acting between the Hilbert spaces H_j and H_i. Secondly, the set of indices of the given R_ij will be more general than banded ones. In fact, we will consider index sets which have an associated graph which is chordal. The case of partial positive operator matrices is also discussed.     

On a class of quasi-Fredholm operators

We study a class of bounded linear operators acting on a Banach spaceX called B-Fredholm operators. Among other things we characterize a B-Fredholm operator as the direct sum of a nilpotent operator and a Fredholm operator and we prove a spectral mapping theorem for B-Fredholm operators.IMemory of my father, Sidi-Bouhouria 1914-0991.     

The perturbation classes problem in Fredholm theory

We give a negative answer to the perturbation classes problem: the perturbation class of the upper semi-Fredholm operators contains properly the strictly singular operators, and the perturbation class of the lower semi-Fredholm operators contains properly the strictly cosingular operators.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .