It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the...

We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence $\mu ={\mu }_n$ of positive numbers and a sequence $f=f_n$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for $f_n\left(T\right)$ is defined by D[f,μ;z](T) = ∑n=0∞ e-μnz fn(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.

We investigate oscillation and spectral properties (sufficient conditions for discreteness and boundedness below of the spectrum) of difference operators B(y)n+k = (-1)nwk n (pk n yk).

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...