The discrete Wiener-Hopf operator generated by a function $a\left(e^{i\theta }\right)$ with the Fourier series ${\sum }_{n\in \mathbb{Z}}{a}_n{e}^{in\theta }$ is the operator T(a) induced by the Toeplitz matrix ${\left(a_{j-k}\right)}_{j,k=0}^{\infty }$ on some weighted sequence space $l^p({\mathbb{Z}}_{+},w)$. We assume that w satisfies the Muckenhoupt $A_p$ condition and that a is a piecewise continuous function subject to some natural multiplier condition. The last condition is in particular satisfied if a is of bounded variation. Our main result is a Fredholm criterion and an index formula for T(a). It implies that the essential spectrum...

We introduce the concept of disjoint hypercyclic operators. These are operators performing the approximation of any given vectors with a common subsequence of iterates applied on a common vector. The notion is extended to sequences of operators, and applied to composition operators and differential operators on spaces of analytic functions.

Let $G$ be a locally compact group and let $1\le p<\infty .$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^p\left(G\right)$ in terms of the weights. Sufficient and...

The purpose of this paper is to show that if σ is the maximal spectral type of Chacon’s transformation, then for any d ≠ d’ we have ${\sigma }^{*d}\perp {\sigma }^{*d^{\text{'}}}$. First, we establish the disjointness of convolutions of the maximal spectral type for the class of dynamical systems that satisfy a certain algebraic condition. Then we show that Chacon’s automorphism belongs to this class.

For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, $({(-\Delta )}^{\alpha }u,\varphi )=(u,{(-\Delta )}^{\alpha }\varphi )$, α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean...

We create a general framework for describing domains of functions of power-bounded operators given by power series with log-convex coefficients. This sheds new light on recent results of Assani, Derriennic, Lin and others. In particular, we resolve an open problem regarding the "one-sided ergodic Hilbert transform" formulated in a 2001 paper by Derriennic and Lin.

After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator $H$. If $H$ is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775,...

2000 Mathematics Subject Classification: 18B30, 47A12.Let A, B be two linear operators on a complex Hilbert space H. We extend a Bouldin's result (1969) conserning W(AB) - the numerical range of the product AB. We show, when AB = BA and A is normal, than W(AB).

In this paper, a method of numerical solution to the dominant eigenvalue problem for positive integral operators is presented. This method is based on results of the theory of positive operators developed by Krein and Rutman. The problem is solved by Monte Carlo method constructing random variables in such a way that differences between results obtained and the exact ones would be arbitrarily small. Some numerical results are shown.

We study conditions under which Dominated Ergodic Theorems hold in rearrangement invariant spaces. Consequences for Orlicz and Lorentz spaces are given. In particular, our results generalize the classical theorems for the spaces $L_p$ and the classes $Llo{g}^{n}L$.

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties....