Some remarks on derivations in Banach algebras and related results

Summary In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = – aD(a ^–1 )a holds for all invertible elementsa A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

Group inverses and Drazin inverses over Banach algebras

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

Some Observations on Riesz Algebras

We give an answer to a question of Huijsmans. In particular the main results of [1, 3] and [5] are generalized. revised 4 February, 11 February, 24 February 2006     

Generalized inverses over Banach algebras

LetA be a matrix over a complex commutative unital Banach algebra. We give necessary and sufficient conditions forA to have a generalized inverse. Moreover, if the Banach algebra has a symmetric involution, these are also necessary and sufficient conditions forA to admit the Moore-Penrose inverse.Partially supported by NSF Grant DMS-8802593     

A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=Nu and the nonlinear time dependent problem u^′=(L+N)u. Under appropriate assumptions on L and N we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.     

Operator norm inequalities in semi-Hilbertian spaces

In this work we extend Cordes inequality, McIntosh inequality and CPR-inequality for the operator seminorm defined by a positive semidefinite bounded linear operator A.     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

Transmission Problems and Boundary Operator Algebras

We examine the operator algebra behind the boundary integral equation method for solving transmission problems. A new type of boundary integral operator, the rotation operator, is introduced, which is more appropriate than operators of double layer type for solving transmission problems for first order elliptic partial differential equations. We give a general invertibility criteria for operators in by defining a Clifford algebra valued Gelfand transform on . The general theory is applied to transmission problems with strongly Lipschitz interfaces for the two classical elliptic operators and . We here use Rellich techniques in a new way to estimate the full complex spectrum of the boundary integral operators. For we use the associated rotation operator to solve the Hilbert boundary value problem and a Riemann type transmission problem. For the Helmholtz equation, we demonstrate how Rellich estimates give an angular spectral estimate on the rotation operator, which with the general spectral mapping properties in translates to a hyperbolic spectral estimate for the double layer potential operator.     

Quasicompact and Riesz endomorphisms of Banach algebras

Let B be a unital commutative semi-simple Banach algebra. We study endomorphisms of B which are also quasicompact operators or Riesz operators. Clearly compact and power compact endomorphisms are Riesz and hence quasicompact. Several general theorems about quasicompact endomorphisms are proved, and these results are then applied to the question of when quasicompact or Riesz endomorphisms of certain algebras are necessarily power compact.     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.     

Banach空间上算子代数的超自反性

In this paper, we introduced the hyperreflexivity of operator algebra A on'Banach space, discussed the necessary, and sufficient condition that A is hyperreflexive, the estimate of hyperreflexive constant and the invariance of hyperreflexivety under the similarity transformation.     

Fredholm theory in ordered Banach algebras

This paper illustrates some initial steps taken in the e?ort of unifying the theory of positivity in ordered Banach algebas (OBAs) with the general Fredholm theory in Banach algebras. We introduce here upper Weyl and upper Browder elements in an OBA relative to an arbitrary Banach algebra homomorphism and investigate the spectra corresponding to the sets of upper Weyl and upper Browder elements, which we shall refer to as the upper Weyl and upper Browder spectra, respectively.     

Gleason Parts and Weakly Compact Homomorphismsbetween Uniform Banach Algebras

 If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis. (Received 9 March 1998; in revised form 29 June 1998)     

Bounded Reflexivity of Operator Spaces

We introduce a concept bounded reflexivity for a subspace of operators on a normed space. We explore the properties of bounded reflexivity, study the similarities and differences between bounded reflexivity and the usual reflexivity for a subspace of operators. As applications of bounded reflexivity, we give alternative proofs of some well known results about positivity and complete positivity of elementary operators.     

Ordered Involutive Operator Spaces

This is a companion to recent papers of the authors; here we construct the ‘noncommutative Shilov boundary’ of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider ‘maximal’ and ‘minimal’ unitizations of such ordered operator spaces.     

On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C^∗-algebra and therefore, derive some new and well-known criteria in operator theory.     

Some Problems on Homomorphisms and Real Function Algebras

 In this paper we solve a problem about the representation of all homomorphisms on a real function algebra as point evaluations and another two about function algebras in which homomorphisms are point evaluations on sequences in the algebra. (Received 4 December 2000; in revised form 2 April 2001)