We consider a discrete Schrödinger operator 𝒥 with Wigner-von Neumann potential not belonging to l². We find the asymptotics of orthonormal polynomials associated to 𝒥. We prove a Weyl-Titchmarsh type formula, which relates the spectral density of 𝒥 to a coefficient in the asymptotics of the orthonormal polynomials.

The Coifman-Fefferman inequality implies quite easily that a Calderón-Zygmund operator T acts boundedly in a Banach lattice X on ℝⁿ if the Hardy-Littlewood maximal operator M is bounded in both X and X'. We establish a converse result under the assumption that X has the Fatou property and X is p-convex and q-concave with some 1 < p, q < ∞: if a linear operator T is bounded in X and T is nondegenerate in a certain sense (for example, if T is a Riesz transform) then M is bounded in both X and...

Let $\left(X_j\right)$ be a sequence of independent identically distributed random operators on a Banach space. We obtain necessary and sufficient conditions for the Abel means of $X_n...X_2{X}_1$ to belong to Hardy and Lipschitz spaces a.s. We also obtain necessary and sufficient conditions on the Fourier coefficients of random Taylor series with bounded martingale coefficients to belong to Lipschitz and Bergman spaces.

We give a brief survey of recent results of order limited operators related to some properties on Banach lattices.

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

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...

A sequence (xn) in a Banach space X is said to be weakly-p-summable, 1 ≤ p &lt; ∞, when for each x* ∈ X*, (x*xn) ∈ lp. We shall say that a sequence (xn) is weakly-p-convergent if for some x ∈ X, (xn - x) is weakly-p-summable.

In this note we review some results about:1. Representation of Absolutely (∞,p) summing operators (∏∞,p) in C(K,E)2. Dunford-Pettis properties.