On product formulas for exponential functions.
On some functions that map each generator of a -semigroup into the generator of a holomorphic semigroup.
On spectral homomorphisms and continuous functions of Banach algebra elements
On the algebra of smooth operators
Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fréchet algebra L(s',s) of so-called smooth operators. We also characterize closed commutative *-subalgebras of L(s',s) and establish a Hölder continuous functional calculus in this algebra. The key tool is the property (DN) of s.
On the differences of the consecutive powers of Banach algebra elements
Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of and .
On the generalized Drazin inverse and generalized resolvent
We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in -algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel....
On the spectral properties of elements of the type exp(ih) in Hermitian Banach algebras
Perturbation theorems for Hermitian elements in Banach algebras
Two well-known theorems for Hermitian elements in C*-algebras are extended to Banach algebras. The first concerns the solution of the equation ax - xb = y, and the second gives sharp bounds for the distance between spectra of a and b when a, b are Hermitian.
Quasi-Banach algebras, ideals, and holomorphic functional calculus
Quotient Banach spaces
Quotient Banach spaces; Multilinear theory
Schur Lemma and the Spectral Mapping Formula
Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula = generates a proper left idealUsing the Schur lemma and the Gelfand-Mazur theorem we prove that has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.
Semigroups and resolvents of bounded variation, imaginary powers and H°° functional calculus.
Several variable spectral theory in the noncommutative case
Sobolev- und Sobolev-Hardy-Räume auf S1: Dualitätstheorie und Funktionalkalküle.
Some simple proofs in holomorphic spectral theory
This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.
Spectra and Numerical Ranges in Locally M-Convex Algebras
Spectral mapping inclusions for the Phillips functional calculus in Banach spaces and algebras
We investigate the weak spectral mapping property (WSMP) , where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators , t ≥ 0, are multipliers.
Spectral pictures of operators quasisimilar to the unilateral shift.
Spectral sets
The paper studies spectral sets of elements of Banach algebras as the zeros of holomorphic functions and describes them in terms of existence of idempotents. A new decomposition theorem characterizing spectral sets is obtained for bounded linear operators.