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 deal with the existence of solutions of the Dirichlet problem for sublinear and superlinear partial differential inclusions considered as generalizations of the Euler-Lagrange equation for a certain integral functional without convexity assumption. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principles which enables approximation of the solution for our problem.

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 show that in contrast to the case of the operator norm topology on the set of regular operators, the Fuglede-Kadison determinant is not continuous on isomorphisms in the group von Neumann algebra $𝒩\left(\mathbb{Z}\right)$ with respect to the strong operator topology. Moreover, in the weak operator topology the determinant is not even continuous on isomorphisms given by multiplication with elements of $\mathbb{Z}\left[\mathbb{Z}\right]$. Finally, we define $T\in 𝒩\left(\mathbb{Z}\right)$ such that for each $\lambda \in ℝ$ the operator $T+\lambda ·{\mathrm{id}}_{l^2\left(\mathbb{Z}\right)}$ is a self-adjoint weak isomorphism of determinant class but...

The Leray-Schauder degree is extended to certain multi-valued mappings on separable Hilbert spaces with applications to the existence of weak periodic solutions of discontinuous semilinear wave equations with fixed ends.

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).

In this article, we consider the operator $L$ defined by the differential expression $$\ell \left(y\right)=-y^{\text{'}\text{'}}+q\left(x\right)y,-\infty <x<\infty$$ in $L_2(-\infty ,\infty )$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition $$\underset{-\infty <x<\infty }{sup}\left\{exp\left(ϵ\sqrt{\left|x\right|}\right)\left|q\left(x\right)\right|\right\}<\infty ,ϵ>0$$ holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.

We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics...

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...

Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are developed for solving the arising systems of convection-diffusion-dispersion-reaction equations, and the received results of several discretization methods are presented. We concentrate on linear reaction systems, which can be solved analytically. In the numerical methods, we use large time-steps to achieve long simulation times of about 10 000 years. We propose...

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...