<Emphasis Type="Italic">C</Emphasis>-Algebras on some Free-Banach Spaces

The main goal of this work is to study the Gelfand spaces of some commutative Banach algebras with unit within the space of bounded linear operators. We will also show, under special condition, that this algebra is isometrically isomorphic to some space of continuous functions defined over a compact. Such isometries preserve idempotent elements. This fact will allow us to define the respective associated measure which is known as spectral measure. Let us also notice that this measure is obtained by restriction of the reciprocal of the Gelfand transform to the set of characteristic functions of clopen subsets of the spectrum of above algebra. We will finish this work showing that each element of such algebras described above can be represented as an integral of some continuous function, where the integral has been defined through the spectral measure.     

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.     

Amenability of algebras of approximable operators

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson’s space.     

Surjective linear maps preserving local spectra

Let B(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize linear surjective and continuous maps on B(X) preserving different local spectral quantities at a nonzero fixed vector.     

Continuity of the Restriction of C 0-Semigroups to Invariant Banach Subspaces

A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for strong continuity of the restriction of a C₀-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators are bounded near the initial time, then the restricted semigroup is strongly continuous.     

An extension of a Phillips's theorem to Banach algebras and application to the uniform continuity of strongly continuous semigroups

In this work we present an extension to arbitrary unital Banach algebras of a result due to Phillips [R.S. Phillips, Spectral theory of semigroups of linear operators, Trans. Amer. Math. Soc. 71 (1951) 393-415] (Theorem 1.1) which provides sufficient conditions assuring the uniform continuity of strongly continuous semigroups of linear operators. It implies that, when dealing with the algebra of bounded operators on a Banach space, the conditions of Phillips's theorem are also necessary. Moreover, it enables us to derive necessary and sufficient conditions in terms of essential spectra which guarantee the uniform continuity of strongly continuous semigroups. We close the paper by discussing the uniform continuity of strongly continuous groups (T(t))_t∈R acting on Banach spaces with separable duals such that, for each t∈R, the essential spectrum of T(t) is a finite set.     

Operators that are points of spectral continuity

In this paper a characterization is obtained of those bounded operators on a Hilbert space at which the spectrum is continuous, where the spectrum is considered as a function whose domain is the set of all operators with the norm topology and whose range is the set of compact subsets of the plane with the Hausdorff metric. Similar characterizations of the points of continuity of the Weyl spectrum, the spectral radius, and the essential spectral radius are also obtained.The first author was supported by National Science Foundation Grant MCS 77-28396.     

An overview of semi-continuity results on the spectral radius and positivity

Our aim is to provide a novelly comprehensive and unifying approach to showing the continuous dependence of the spectral radius of compact linear operators defined on Banach spaces (with specific attention to positive operators defined on normal Banach spaces) and emphasizing that the upper semi-continuity generally holds unlike the lower semi-continuity.     

Weighted composition operators and locally convex algebras

The Gleason-Kahane-Zelazko theorem characterizes the continuous homomorphism of an associative, locally multiplicatively convex, sequentially complete algebra A into the field ? among all linear forms on A. This characterization will be applied along two different directions. In the case in which A is a commutative Banach algebra, the theorem yields the representation of some classes of continuous linear maps A : A→ A as weighted composition operators, or as composition operators when A is a continuous algebra endomorphism. The theorem will then be applied to explore the behaviour of continuous linear forms on quasi-regular elements, when A is either the algebra of all Hilbert-Schmidt operators or a Hilbert algebra.     

Weighted composition operators and locally convex algebras

The Gleason-Kahane-Zelazko theorem characterizes the continuous homo-morphism of an associative, locally multiplicatively convex, sequentially complete algebra A into the field C among all linear forms on A. This characterization will be applied along two different directions. In the case in which A is a commutative Banach algebra, the theorem yields the representation of some classes of continuous linear maps A:A→A as weighted composition operators, or as composition operators when A is a continuous algebra endomorphism. The theorem will then be applied to explore the behaviour of continuous linear forms on quasi-regular elements, when A is either the algebra of all Hilbert-Schmidt operators or a Hilbert algebra.     

K-POTENT PRESERVING LINEAR MAPS

Let B(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let k ≥ 2 be an integer and φ a weakly continuous linear surjective map from B(X) into itself. It is shown that φ is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number λ satisfying λk-1= 1. Let A be a von Neumann algebra and B be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multiplied by an invertible element with (k - l)-th power I.     

Aspects of operator theory in hereditarily indecomposable Banach spaces

We examine certain special features exhibited by various classes of linear operators acting in a hereditarily indecomposable Banach space. For instance, we show that the family of all Riesz operators in a H.I. space forms a closed, 2-sided ideal. We also give further characterizations of the class of scalar-type spectral operators (to those already given in [16]). The final section discusses some properties of the spectral maximal spaces of (necessarily decomposable) linear operators in such spaces. Conferenza tenuta il 16 settembre 1997 The support of the German Academic Exchange Scheme (DAAD) is gratefully acknowledged     

The generalised Berger–Wang formula and the spectral radius of linear cocycles

Using ergodic theory we prove two formulae describing the relationships between different notions of joint spectral radius for sets of bounded linear operators acting on a Banach space. The first formula was previously obtained by V.S. Shulman and Yu.V. Turovski? using operator-theoretic ideas. The second formula shows that the joint spectral radii corresponding to several standard measures of noncompactness share a common value when applied to a given precompact set of operators. This result may be seen as an extension of classical formulae for the essential spectral radius given by R. Nussbaum, A. Lebow and M. Schechter. Both results are obtained as a consequence of a more general theorem concerned with continuous operator cocycles defined over a compact dynamical system. As a byproduct of our method we answer a question of J.E. Cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle.     

Universal and chaotic multipliers on spaces of operators

We use tensor product techniques to study universality, hypercyclicity and chaos of multipliers defined on operator ideals and of multiplication operators on the space of all continuous and linear operators, thus continuing the work of Kit Chan. We also obtain the first examples of outer multipliers on a Banach algebra which are chaotic in the sense of Devaney, and prove sufficient conditions for the existence of closed subspaces of universal vectors for operators between Fréchet spaces.     

The σ<Emphasis Type="Italic">g</Emphasis>-Drazin Inverse and the Generalized Mbekhta Decomposition

In this paper we define and study an extension of the g-Drazin for elements of a Banach algebra and for bounded linear operators based on an isolated spectral set rather than on an isolated spectral point. We investigate salient properties of the new inverse and its continuity, and illustrate its usefulness with an application to differential equations. Generalized Mbekhta subspaces are introduced and the corresponding extended Mbekhta decomposition gives a characterization of circularly isolated spectral sets.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

Extending the formula to calculate the spectral radius of an operator

In a Banach space, Gelfand's formula is used to find the spectral radius of a continuous linear operator. In this paper, we show another way to find the spectral radius of a bounded linear operator in a complete topological linear space. We also show that Gelfand's formula holds in a more general setting if we generalize the definition of the norm for a bounded linear operator.

     

On uniform continuity of the spectral radius in Banach algebras

We investigate relations between different forms of subadditivity and submultiplicativity of the spectral radius. In particular, we prove that if the spectral radius is uniformly continuous on a Banach algebra, then the algebra is commutative modulo the radical; this confirms a conjecture raised by the second author in [7].     

Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

We present here a quite unexpected result: Apart from already known commutative C*-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.