Mostafa Mbekhta

Linear maps preserving the set of Fredholm operators

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper we characterize surjective linear maps Φ: FB(H)→ B(H) preserving the set of Fredholm operators in both directions. As an application we prove that Φ preserves the essential spectrum if and only if the ideal of all compact operators is invariant under Φ and the induced linear map φ on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either ind (Φ)(T)) = ind(T) or ind(Φ (T)) = -ind(T) for every Fredholm operator T.

Linear maps preserving semi-Fredholm operators and generalized invertibility

Let H be an infinite-dimensional separable complex Hilbert space and the algebra of all bounded linear operators on H. We characterize surjective linear maps preserving semi-Fredholm operators in both directions. As an application we substantially improve a recently obtained characterization of linear preservers of generalized invertibility (Mbekhta, M., Rodman, L. and Šemrl, P., 2006, Linear maps preserving generalized invertibility, Integral Equations Operator Theory, 55, 93–109.). The new proof given in this note is not only more efficient, but also much shorter and simpler.

Local spectrum and generalized spectrum

This paper provides the proofs of those results announced in [5, ?5] that deal with the connection between the regular set and the local resolvant set of closed operators on a Hilbert space. We also give some characterizations and properties of Cowen-Douglas operators.

Generalized inverses and the maximal radius of regularity of a Fredholm operator

Operators possessing analytic generalized inverses satisfying the resolvent identity are studied. Several characterizations and necessary conditions are obtained. The maximal radius of regularity for a Fredholm operatorT is computed in terms of the spectral radius of a generalized inverse ofT. This provides a partial answer to a conjecture of J. Zemánek.

Operators with finite chain length and the ergodic theorem

With a technical assumption (E-k), which is a relaxed version of the condition Tn/n O 0, n -n o0, where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition (X = (I T)kX D ker(I T)k), of closedness of (I T)kX, and of quasi-Fredholmness of I T. One consequence, still assuming (E-k), is that I T is semi-Fredholm if and only if I T is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras. 1. OPERATORS WITH FINITE CHAINS We begin with a few algebraic observations. Let S be a linear operator on the vector space X. If there is an integer n for which SnX = Sn+1X, then we say that S has finite descent and the smallest integer d(S) for which this equality occurs is called the descent of S. If there is an integer m for which ker Sm = kerSm+ , then S is said to have finite ascent and the smallest integer a(S) for which this equality occurs is called the ascent of S. If both a(S) and d(S) are finite, then they are equal [3], 38.3; we say that S is chain-finite and that its chain length is this common minimal value. Moreover ([3], 38.4), in this case there is a decomposition of the vector space

On the commutant and orbits of conjugation

In this note, we analyse the relationship between the commutant of a bounded linear operator A and the algebra of similarity B A that was introduced in the late 70s as a characterization of nest algebras. Necessary and sufficient conditions are also obtained for an operator to commute with real scalar generalized operators in the sense of Colojoara-Foias in Banach spaces. In the second part, we analyse the relationship between the generalized inverse, the generalized commutant and the orbits of conjugation.

On the isolated points of the surjective spectrum of a bounded operator

For a bounded operator T acting on a complex Banach space, we show that if T - A is not surjective, then λ is an isolated point of the surjective spectrum σ su (T) of T if and only if X = H 0 (T-λ)+K(T-λ), where H 0 (T) is the quasinilpotent part of T and K(T) is the analytic core for T. Moreover, we study the operators for which dim K(T) < oo. We show that for each of these operators T, there exists a finite set E consisting of Riesz points for T such that 0 ∈ σ(T) \ E and σ(T) \ E is connected, and derive some consequences.

Operators with bounded conjugation orbits

For a bounded invertible operator A on a complex Banach space X, letBA be the set of operators T in L(X) for which supn≥0 ‖AnTA−n‖ <∞. Suppose that Sp(A) = {1} and T is in BA ∩ BA−1 . A bound is given on ‖ATA−1 − T‖ in terms of the spectral radius of the commutator. Replacing the condition T in BA−1 by the weaker condition ‖A−nTAn‖ = o(e √ n), as n→∞ for every > 0, an extension of the Deddens-Stampfli-Williams results on the commutant of A is given.

Partial isometries: Factorization and connected components

In this paper we compute the distance between the connected components of the sets of all partial isometries and essential partial isometries. Some factorization results are also proved. This complements previous results due to P.R. Halmos and J.E. McLaughlin

Ascente, descente et spectre essentiel quasi-Fredholm

RésuméDans ce travail, nous donnons quelques propriétés du spectre essentiel quasi-Fredholm d’un opérateur fermé dans un Hilbert. Par ailleurs, nous montrons que l’intersection de l’ensemble des points quasi-Fredholm avec l’ensemble de Riesz d’un opérateur est un ouvert de C, inclus dans l’ensemble résolvant (sauf éventuellement un ensemble dénombrable de points qui sont en fait des poles d’ordre fini de l’opérateur résolvant). D’autre part, nous étudions les points quasi-Fredholm situés dans la frontière du spectre et nous montrons que ces points sont de Riesz. Enfin, nous donnons plusieurs caractérisations des opérateurs dont le spectre essentiel quasi-Fredholm est vide.