The essential spectrum of bundle shifts over Parreau-Widom domains is studied. Such shifts are models for subnormal operators of special (Hardy) type considered earlier in [AD], [R1] and [R2]. By relating a subnormal operator to the fiber of the maximal ideal space, an application to cluster values of bounded analytic functions is obtained.

Let $E$ be a complex Banach space, with the unit ball $B_E$. We study the spectrum of a bounded weighted composition operator $u{C}_{\varphi }$ on $H^{\infty }\left(B_E\right)$ determined by an analytic symbol $\varphi$ with a fixed point in $B_E$ such that $\varphi \left(B_E\right)$ is a relatively compact subset of $E$, where $u$ is an analytic function on $B_E$.

Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk ${}^N$ or on the Segal-Bargmann space over ${ℂ}^N$. Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression $A_n$ of A to the linear span of the monomials $z_1^{k_1}...z_N^{k_N}:0\le k_j\le n$. Unfortunately, in general the spectrum of $A_n$ does not mimic the spectrum of A as...

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.

We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂̅-problems with certain estimates can be solved. We show that the projection of the spectrum onto ℂⁿ equals Ω̅ and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.

Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$. Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d=2$ or $3$, we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$); for $d=2$, we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem $3$).

Let HE∞ be the space of all bounded holomorphic functions on the unit ball of the Banach space E. In this note we study the algebra homomorphisms on HE∞ which are strict continuous.

Let $A_{\zeta }=\Omega -\overline{\rho \left(\zeta \right)·\Omega }$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{\zeta }\left(z\right)$ of $A_{\zeta }$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{\zeta }$ is non-pseudoconvex when the dimension of $A_{\zeta }$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ²log{K}_{\zeta }/\partial \zeta \partial \zeta ̅$, as well as its boundary behavior.

The notion of “strong boundary values” was introduced by the authors in the local theory of hyperfunction boundary values (boundary values of functions with unrestricted growth, not necessarily solutions of a PDE). In this paper two points are clarified, at least in the global setting (compact boundaries): independence with respect to the defining function that defines the boundary, and the spaces of test functions to be used. The proofs rely crucially on simple results in spectral asymptotics.

Let E be a Frechet (resp. Frechet-Hilbert) space. It is shown that E ∈ (Ω) (resp. E ∈ (DN)) if and only if [H(OE)]' ∈ (Ω) (resp. [H(OE)]' ∈ (DN)). Moreover it is also shown that E ∈ (DN) if and only if Hb(E') ∈ (DN). In the nuclear case these results were proved by Meise and Vogt [2].

Polynomials on ${ℝ}^n$ with values in an irreducible ${Spin}_n$-module form a natural representation space for the group ${Spin}_n$. These representations are completely reducible. In the paper, we give a complete description of their decompositions into irreducible components for polynomials with values in a certain range of irreducible modules. The results are used to describe the structure of kernels of conformally invariant elliptic first order systems acting on maps on ${ℝ}^n$ with values in these modules.

General topological conditions are given for integration cycles of a certain class of integral formulas for holomorphic functions of several complex variables.