In this paper we suggest a general framework of the spectral mapping theorem in terms of parametrized Banach space bicomplexes.

We investigate the weak spectral mapping property (WSMP) $\overline{\mu ̂\left(\sigma \right(A\left)\right)}=\sigma \left(\mu ̂\left(A\right)\right)$, 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 $e^{At}$, t ≥ 0, are multipliers.

Let M be a Beurling-type submodule of $H²{(}_d)$, the Hardy space over the unit ball ${}_d$ of ${ℂ}^d$, and let $N=H²{(}_d)/M$ be the associated quotient module. We completely describe the spectrum and essential spectrum of N, and related index theory.

We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au\left(x\right)=\left({\sum }_{k=1}^N{e}^{{\phi }_k}{T}_{{\alpha }_k}\right)u\left(x\right)={\sum }_{k=1}^N{e}^{{\phi }_k\left(x\right)}u\left({\alpha }_k\left(x\right)\right)$, u ∈ C(X), where ${\phi }_k\in C\left(X\right)$ and ${\alpha }_k:X\to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $\phi ={\left({\phi }_k\right)}_{k=1}^N$. We prove that $ln\left(r\left(A\right)\right)=\lambda \left(\phi \right)=ma{x}_{\nu \in Mes}{\sum }_{k=1}^N{\int }_X{\phi }_{k}d{\nu }_k-\lambda *\left(\nu \right)$, where Mes is the set of all probability vectors of measures $\nu ={\left({\nu }_k\right)}_{k=1}^N$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on $\left(C^N\left(X\right)\right)*$. In other...

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.

This paper studies properties of a large class of algebras of holomorphic functions with bounded growth in several complex variables.The main result is useful in the applications. Using the symbolic calculus of L. Waelbroeck, it gives for instance a theorem of the “Nullstellensatz” type and approximation theorems.

Let G be a locally compact abelian group and ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. We study Hilbert transforms associated with G-flows on ℳ and closed semigroups Σ of Ĝ satisfying the condition Σ ∪ (-Σ) = Ĝ. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from L¹(ℳ,τ) into $L^{1,\infty }(ℳ,\tau )$. As an application, we obtain a Matsaev-type result for p = 1: if x is a quasi-nilpotent compact operator...

In this note we define and explore, à la Godement, spectral subspaces of Banach space representations of the Fourier-Eymard algebra of a (nonabelian) locally compact group.

It is shown that every closed rotation and translation invariant subspace $V$ of $C\left({\mathbf{R}}^n\right)$ or $\delta \left({\mathbf{R}}^n\right)$, $n\ge 2$, is of spectral synthesis, i.e. $V$ is spanned by the polynomial-exponential functions it contains. It is a classical problem to find those measures $\mu$ of compact support on ${\mathbf{R}}^2$ with the following property: (P) The only function $f\in C\left({\mathbf{R}}^2\right)$ satisfying ${\int }_{{\mathbf{R}}^2}f\circ \sigma d\mu =0$ for all rigid motions $\sigma$ of ${\mathbf{R}}^2$ is the zero function. As an application of the above result a characterization of such measures is obtained in terms of their Fourier-Laplace transforms....

On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $su{p}_{n\in \mathbb{N},z\in }\left|\right|{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))k^{-1}{z}^k{U}^k\left|\right|<\infty$. (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with Young-Stieltjes...