Nous étudions les sous-espaces biinvariants du shift usuel sur les espaces à poids$$L_{\omega }^2=\left\{f\in L^2\left(𝕋\right):\parallel f{\parallel }_{\omega }={\left(\sum _{n\in \mathbb{Z}}\left|f\left(n\right)\right|{\omega }^2\left(n\right)\right)}^{1/2}\<+\infty \right\},$$où $\omega \left(n\right)=(1+n{)}^p,n\ge 0$ et $\frac{\omega \left(n\right)}{(1+|n|{)}^p}\overrightarrow{n\to -\infty }+\infty$, pour un certain entier $p\ge 1$. Nous montrons que la trace analytique de tout sous-espace biinvariant est de type spectral, lorsque ${\sum }_{n\ge 2}\frac{1}{nlog\omega (-n)}$ diverge, mais que ceci n’est plus valable lorsque ${\sum }_{n\ge 2}\frac{1}{nlog\omega (-n)}$ converge.

A simple theorem is proved which states a sufficient condition for the sum ot two closed subspaces of a Banach space to be closed. This leads to several analogues of Sarason’s theorem which states that $H^{\infty }+C$ is a closed subalgebra of $L^{\infty }$. In these analogues, the unit circle is replaces by other groups, and the unit disc is replaced by polydiscs or by balls in spaces of several complex variables. Sums of closed ideals in Banach algebras are also studied.

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.

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.

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 A be a semisimple commutative regular tauberian Banach algebra with spectrum ${\Sigma }_A$. In this paper, we study the norm spectra of elements of $\overline{span}{\Sigma }_A$ and present some applications. In particular, we characterize the discreteness of ${\Sigma }_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $\phi \in \overline{}{\Sigma }_{A}0$, φ has a nonempty norm spectrum. For a locally compact group G, let ${ℳ}_2^d\left(Ĝ\right)$ denote the C*-algebra generated by left translation operators on $L^2\left(G\right)$ and $G_d$ denote the discrete group G. We prove that the Fourier...