A spectral criterion of asymptotic almost periodicity for uniformly continuous representations of abelian semigroups

Assume that the unitary asymptotic spectrum of a representation T is at most countable. Then for asymptotic almost periodicity it is necessary and sufficient that the eigensubspaces of the representations T, T^*, corresponding to the unitary weights, should be in duality.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 38–43, 1988.     

Asymptotic almost periodicity and motions of semigroups of operators

In this survey we present an overview of recent work in resolving certain questions that arise in a qualitative study of solutions to the abstract Cauchy problem. Focusing on motions of C₀-semigroups of operators and their asymptotic behavior, we address (1) the characterization of asymptotically almost periodic motions and (2) the characterization of those asymptotically almost periodic functions for which this property carries over to their integral. Moreover, we take up the problem raised by Nemytskii and Stepanov as to when the ω-limit set of a motion of a dynamical system is a minimal set of almost periodic motions.     

Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups

We show that if (S(t)) _t≧0 is a contraction semigroup on a closed convex subset of a uniformly convex Banach space, then every bounded and “asymptotically isometric” almost-orbit of (S(t)) _t≧0 is weakly almost periodic in the sense of Eberlein. As one consequence, results on the existence of almost periodic solutions to the abstract Cauchy problem are obtained without the need fora priori compactness assumptions. As a further consequence, the known strong ergodic limit theorems for (almost-) orbits of nonlinear contraction semigroups turn out to be special cases of Eberlein’s classical ergodic theorem for weakly almost periodic functions.     

Stability and almost periodicity of asymptotically dominated semigroups of positive operators

We discuss conditions such that strong stability and strong asymptotic compactness of a (discrete or continuous) semiflow defined on a subset in the positive cone of an ordered Banach space is preserved under asymptotic domination. This is used to show that on a Banach lattice with order continuous norm strong stability and almost periodicity of a (discrete or strongly continuous) semigroup of positive operators is preserved under asymptotic domination.

     

A spectral mapping theorem for representations of one-parameter groups

In this paper we present some generalization (at the same time a new and a short proof in the Banach algebra context) of the Weak Spectral Mapping Theorem (WSMT) for non-quasianalytic representations of one-parameter groups.

     

Norm-ideal perturbations of one-parameter semigroups and applications

The notion of equivalence classes of generators of one-parameter semigroups based on the convergence of the Dyson expansion can be traced back to the seminal work of Hille and Phillips, who in Chapter XIII of the 1957 edition of their Functional Analysis monograph, developed the theory in minute detail. Following their approach of regarding the Dyson expansion as a central object, in the first part of this paper we examine a general framework for perturbation of generators relative to the Schatten-von Neumann ideals on Hilbert spaces. This allows us to develop a graded family of equivalence relations on generators, which refine the classical notion and provide stronger-than-expected properties of convergence for the tail of the perturbation series. We then show how this framework realises in the context of non-self-adjoint Schrödinger operators.     

Asymptotic behavior of one-parameter semigroups of positive operators

In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.     

Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups

We consider the Mosco convergence of the sets of fixed points for one-parameter strongly continuous semigroups of nonexpansive mappings. One of our main results is the following: Let C$C$ be a closed convex subset of a Hilbert space E$E$. Let {T(t):t≥0}$\{T\left(t\right):t\ge 0\}$ be a strongly continuous semigroup of nonexpansive mappings on C$C$. The set of all fixed points of T(t)$T\left(t\right)$ is denoted by F(T(t))$F\left(T\left(t\right)\right)$ for each t≥0$t\ge 0$. Let τ$\tau$ be a nonnegative real number and let {_tn}$\left\{t_n\right\}$ be a sequence in R$\mathbb{R}$ satisfying τ+_tn≥0$\tau +t_n\ge 0$ and _tn≠0$t_n\ne 0$ for n∈N$n\in \mathbb{N}$, and _limntn=0${lim}_n{t}_n=0$. Then {F(T(τ+_tn))}$\left\{F\left(T(\tau +t_n)\right)\right\}$ converges to _?t≥0F(T(t))${?}_{t\ge 0}F\left(T\left(t\right)\right)$ in the sense of Mosco.     

A class of semigroups having almost trivial multiplications

In this note we completely describe the structure of algebraic semigroups S with Sx=S or Sx degenerate and xS=S or xS degenerate for each x∈S. We then apply our results in characterizing the separately continuous multiplications on a topological space whose only self-maps are either surjective or have degenerate image. In particular, we find that any locally compact Hausdorff space with this property can admit only trivial separately continuous multiplications. Examples of spaces satisfying this property are certain continua discovered by H. Cook [1]. The authors are indebted to Karl H. Hofmann and James T. Rogers for conversations elucidating the topological implications of Theorem 1. The second author supported by NSF Grant GP28655     

On almost periodicity of morphic sequences

It is proved that the problem of determining whether a given morphic sequence is almost periodic is decidable.     

An analyticity criterion for regularized semigroups

A generalization of Kato's analyticity criterion for -semigroups to exponentially bounded regularized semigroups is given by using the method of Laplace transforms.

     

On Liu's analyticity criterion for semigroups

A contraction version of Liu's recent analyticity criterion for semigroups is given a very simple proof based on the exponential formula.     

On the spectral mapping theorem for one-parameter groups of operators

Let A be the generator of a strongly continuous nonquasianatytic oneparameter group of operators U(t) (A, in general, is unbounded). In this case one proves a spectral mapping theorem in the following form:. The proof is based on the theory of operators with separable spectrum.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 146–150, 1989.     

A criterion for a continuous spectral density

For a given weakly stationary random field indexed by the integer lattice of an arbitrary finite dimension, a necessary and sufficient condition is given for the existence of a continuous spectral density. The condition involves the covariances of pairs of sums of the random variables, with the two index sets being “separated” from each other (but possibly “interlaced”) by a certain distance along a coordinate direction.     

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier, under some additional condition (the position of the Denjoy–Wolff point) it was shown in [13] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy–Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy–Wolff point (the case not covered in [13]) than with all the other boundary fixed points of the semigroup.     

Weak almost periodicity of haar measurable functions

LetG be a locally compact group. A weakened version of Grothendieck's double limit criterion [6; p.183] is shown to characterize those2112 (G) that are locally almost everywhere equal to acontinuous weakly almost periodic function in the sense of Eberlein. Additional measure theoretic conditions guarantee continuity of such . As a by-product, we obtain a short proof of the classical result that is continuous when almost periodic.