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Operational quantities derived from the norm and generalized Fredholm theory

Manuel Gonzalez, Antonio Martinón (1991)

Commentationes Mathematicae Universitatis Carolinae

We introduce and study some operational quantities associated to a space ideal 𝔸 . These quantities are used to define generalized semi-Fredholm operators associated to 𝔸 , and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that the classes of generalized semi-Fredholm operators are non-trivial for several classical space ideals.

Operator fractional-linear transformations: convexity and compactness of image; applications

V. Khatskevich, V. Shul'Man (1995)

Studia Mathematica

The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space ℒ ( X 1 , X 2 ) of all linear bounded operators acting from X 1 into X 2 , where X 1 , X 2 are Banach spaces. We show that in the case of Hilbert spaces X 1 , X 2 the image F(ℬ) of any (open or closed) ball ℬ ⊂ D(F) is convex, and if ℬ is closed, then F(ℬ) is compact in the weak operator topology (w.o.t.) (Theorem 1.2). These results extend the corresponding results on compactness obtained in [3],...

Operator semigroups in Banach space theory

Pietro Aiena, Manuel González, Antonio Martínez-Abejón (2001)

Bollettino dell'Unione Matematica Italiana

In questo lavoro, motivati dalla teoria di Fredholm in spazi di Banach e dalla cosiddetta teoria degli ideali di operatori nel senso di Pietsch, viene definito un nuovo concetto di semigruppo di operatori. Questa nuova definizione include quella di molte classi di operatori già studiate in letteratura, come la classe degli operatori di semi-Fredholm, quella degli operatori tauberiani ed altre ancora. Inoltre permette un nuovo ed unificante approccio ad una serie di problemi in teoria degli operatori...

Perturbation theory relative to a Banach algebra of operators

Bruce Barnes (1993)

Studia Mathematica

Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to ℬ of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to ℬ is studied. Several examples are given to illustrate the usefulness of these concepts.

Polaroid type operators and compact perturbations

Chun Guang Li, Ting Ting Zhou (2014)

Studia Mathematica

A bounded linear operator T acting on a Hilbert space is said to be polaroid if each isolated point in the spectrum is a pole of the resolvent of T. There are several generalizations of the polaroid property. We investigate compact perturbations of polaroid type operators. We prove that, given an operator T and ε > 0, there exists a compact operator K with ||K|| < ε such that T + K is polaroid. Moreover, we characterize those operators for which a certain polaroid type property is stable under...

Polaroid type operators under perturbations

Pietro Aiena, Elvis Aponte (2013)

Studia Mathematica

A bounded operator T defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. The "polaroid" condition is related to the conditions of being left polaroid, right polaroid, or a-polaroid. In this paper we explore all these conditions under commuting perturbations K. As a consequence, we give a general framework from which we obtain, and also extend, recent results concerning Weyl type theorems (generalized or not) for T + K, where K is an...

Quotients of L1 by reflexive subspaces.

Manuel González, Antonio Martínez-Abejón (1997)

Extracta Mathematicae

Here we present and example and some results suggesting that there is no infinite-dimensional reflexive subspace Z of L1 ≡ L1[0,1] such that the quotient L1/Z is isomorphic to a subspace of L1.

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