The main object of this paper is to introduce and study some sequence spaces which arise from the notation of generalized de la Vallée–Poussin means and the concept of a modulus function.

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.

T. Trent gave a new characterization of subnormality for an operator on a Hilbert space. T. Bînzar and D. Păunescu generalized this condition to commuting triples of operators. Here, we give an n-variable unbounded version of the above results. Theorems of this kind have also been obtained by Z. J. Jabłoński and J. Stochel.

Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^p\left(I\right)$, 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$ and their preduals.

For the difference equation $\left(ϵ\right)x_{n+1}=A{x}_n+ϵ{\sum }_{k=-\infty }^n{R}_{n-k}{x}_k$,where $x_n\in Y,Y$  is a Banach space, $ϵ$ is a parameter and  $A$  is a linear, bounded operator. A sufficient condition for the existence of a unique special solution  $y={\left\{y_n\right\}}_{n=-\infty }^{\infty }$  passing through the point  $x_0\in Y$  is proved. This special solution converges to the solution of the equation (0) as  $ϵ\to 0$.

Toeplitz operators on strongly pseudoconvex domains in Cn, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on Lp, as the power of the distance varies, are established.

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.

We give a simple proof of the relation between the spectra of the difference and product of any two idempotents in a Banach algebra. We also give the relation between the spectra of their sum and product.