Ernst Albrecht

Some Topics in the Theory of Decomposable Operators

A bounded linear operator T on a Banach space X is called decomposable if for every open covering {Ω1, Ω2 } of the complex plane C there are closed invariant subspaces X1, X2 for T such that sp(T,Xj)⊂Ω j for j = 1,2 and X = X1 + X2 (where sp(T,Xj) is the spectrum of T on Xj). We refer to the monographs [7, 16] for the theory of decomposable operators. For bounded linear operators T and S on Banach spaces X resp.

Stability of the index of a semi-Fredholm complex of Banach spaces

Abstract We prove the stability of the index and the semicontinuity of the dimensions of the cohomology groups of semi-Fredholm complexes of Banach spaces and closed linear operators with respect to perturbations of the operators and of the underlying spaces which are small with respect to the gap topologies. It seems that, even for single semi-Fredholm operators, some of the statements are more general than the current ones. The results are applied to obtain semicontinuity for joint spectra of finite systems of commuting bounded linear operators.

Decomposable Systems of Operators in Harmonic Analysis

Recently (see [2]), we introduced the notion of decomposability for arbitrary (not necessarily finite) systems of commuting bounded linear operators on a Banach space and extended several results of I. Colojoară, C. Foias, Şt. Frunză and the author to this general situation. We now apply this theory to multiplication operators on Banach algebras and study multipliers on Lp (G), 1≤p<∞ for locally compact abelian groups G.

Automatic Continuity of Generalized Local Linear Operators.

In this note, we present a general automatic continuity theory for linear mappings between certain topological vector spaces. The theory applies, in particular, to local operators between spaces of functions and distributions, to algebraic homomorphisms between certain topological algebras, and to linear mappings intertwining generalized scalar operators.

Funktionalkalküle in mehreren Veränderlichen für stetige lineare Operatoren auf Banachräumen

Very important results and tools in the theory of generalized scalar operators are the theorem of support of C. Foias [10] (see also [8], Th. 3.1.6) and the theorem that every generalized scalar operator is decomposable ([8], Th. 3.1.19). This note contains results of this type for functional calculi in several variables. Moreover, we give (as in [15] in the case of one variable) a characterization of spectral distributions with support contained in ℝn resp. in Γn (where Г={z∈ℂ:|z|=1}).

On two questions of I. Colojoară and C. Foiaş

An example of a decomposable operator is given which is not strongly decomposable. This answers two questions (see (Q1) and (Q2)) of I. Colojoară and C. Foiaş (in [5], 6.5. (b), (c)) in the negative.

Non-analytic functional calculi in several variables

Generalizing a result of [1] and [2], we show that every-scalar system (see Def. 3) is decomposable (in the sense of [6], [7]). By means of this fact and by some results on decomposable n-tuples (contained in [6] and [7]) we prove the theorem of support and (in the case of an inverse closed admissible algebra) the spectral mapping theorem for-functional calculi in several variables (see Def. 3). In the case of a single operator we obtain a simplification of the definition of an-spectral function (in the sense of [5], Def. 3.1.3).

Local operators between spaces of ultradifferentiable functions and ultradistributions

In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.

Multicyclic Systems of Commuting Operators

In this note we establish multidimensional variants of some of the analytic structure results of D.A. Herrero [6], [8], [9], [10] (see also Chapter 11 of the monograph [2]). Roughly speaking, Herrero proved among other statements that n-multicyclic operators on an infinite dimensional (complex) Banach space X with the additional property that the set