Fredholm composition operators

Fredholm composition operators on a variety of Hilbert spaces of analytic functions on domains in , are characterized.

     

Affine mappings of invertible operators

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.

     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).     

Perturbations of Drazin invertible operators

The necessary and sufficient conditions for the small norm perturbation of a Drazin invertible operator to be still Drazin invertible and the sufficient conditions for the finite rank perturbation of a Drazin invertible operator to be still Drazin invertible are established.     

A solution to a problem on invertible disjointness preserving operators

We construct an invertible disjointness preserving operator on a normed lattice such that is not disjointness preserving.

     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

Fredholm property of non-smooth pseudodifferential operators

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator P which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.     

Fredholm operators on locally compact spaces

Sunto Si definiscono operatori localmente ellittici su varietà topologiche; per ogni coppia di operatori localmente ellittici, coincidenti al di fuori di un compatto, si introduce un indice (analitico) di Fredholm. Queste tecniche permettono di estendere i risultati di D. Sullivan e dell'autore [S-T], [T] alla categoria delle varietà topologiche localmente compatte.

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Dedicato al Professor Enzo Martinelli in segno dell'amicizia di cui mi onora

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Research supported by the N.S.F. Grant N MCS 8102758. This paper was written at State University of New York, Center at Stony Brook (June 1982).     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Linear boundary-value problems described by Drazin invertible operators

Mathematical Notes - The main subject of this paper is the study of a general linear boundary-value problem with Drazin or right Drazin (respectively, left Drazin) invertible operators...     

Multiparameter perturbation theory of Fredholm operators applied to bloch functions

In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set Λ of parameters t = (t ₁, ..., t _n ) for which the equation A(t)u = 0 has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set Λ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set E(k) in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.     

Generalized super Gabor duals with bounded invertible operators

In this paper, we introduce generalized super Gabor duals with bounded invertible operators by combining ideas concerning super Gabor frames with the idea of g-duals as proposed by Dehgham and Fard in 2013. Given a super Gabor frame and a bounded invertible operator A, we characterize its generalized super Gabor duals with A, and derive a parametric expression of all its generalized super Gabor duals with A. The perturbation of generalized super Gabor duals is considered as well.