We improve (in some sense) a recent theorem due to Banas and Knap (1989) about the existence of integrable solutions of a functional-integral equation.

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1-\Phi \left(z\right)$ inside the unit disk, where $\Phi$ is the generating function of $X$, under some mild conditions

We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBab}\right)$, $\left(\mathrm{SBw}\right)$ and $\left(\mathrm{SBb}\right)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $\left(\mathrm{SBab}\right)$, then $S\oplus T$ has the property $\left(\mathrm{SBab}\right)$ if and only if ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}(S\oplus T)={\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(S\right)\cup {\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$, where ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBb}\right)$ and $\left(\mathrm{SBw}\right)$ with...

On donne dans cet exposé des bornes inférieures universelles, en limite semiclassique, de la hauteur des résonances de forme associées aux opérateurs de Schrödinger à l’extérieur d’obstacles avec des conditions au bord de Dirichlet ou de Neumann et des potentiels analytiquement dilatables et tendant vers $0$ à l’infini. Ces bornes inférieures sont exponentiellement petites par rapport à la constante de Planck.

This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights $\lambda ₙ:=n^{\alpha }(1+\Delta ₙ)$ where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and $\lambda ₙ=\surd \left(q_{n-1}qₙ\right)$.

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X,|\left|·\right|{|}_X)$ and $(Y,|\left|·\right|{|}_Y)$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $\left|\right|T\left(1_{Aₙ}f\right){\left|\right|}_Y\to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

We study the discrete Schrödinger operator $H$ in ${𝐙}^d$ with the surface quasi periodic potential $V\left(x\right)=g\delta \left(x_1\right)tan\pi (\alpha ·x_2+\omega )$, where $x=(x_1,x_2),x_1\in {𝐙}^{d_1},x_2\in {𝐙}^{d_2},\alpha \in {𝐑}^{d_2},\omega \in [0,1)$. We first discuss a proof of the pure absolute continuity of the spectrum of $H$ on the interval $[-d,d]$ (the spectrum of the discrete laplacian) in the case where the components of $\alpha$ are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane ${𝐙}^{d_1}$ waves, the form...