A Hille-Yoshida-Phillips theorem for discrete semigroups on complete ultrametric locally convex spaces (original) (raw)

Demicompactness Results for Strongly Continuous Semigroups, Generators and Resolvents

Mediterranean Journal of Mathematics, 2018

Let (U (t)) t≥0 be a strongly continuous semigroup of bounded linear operators on a Banach space X and B be a bounded operator on X. In this paper, we develop some aspects of the theory of semigroup for which U (t)B (respectively, BU (t), BU (t)B) is demicompact for some (respectively, every) t > 0. In addition, we study the demicompactness of similar, subspace and product semigroups. We also investigate the demicompactness of the resolvent. We close this paper by giving some conditions guaranteeing the demicompactness of a generator of a strongly continuous semigroup in a Hilbert space.

Boundedness properties of resolvents and semigroups of operators

Banach Center Publications, 1997

T j x 2 ≤ M (T) 2 x 2 is satisfied. Also suppose that the adjoint T * of the operator T is square bounded in average with constant M (T *). Then the operator T is power bounded in the sense that sup{ T n : n ∈ N} is finite. In fact the following inequality is valid for all n ∈ N: T n ≤ eM (T)M (T *). Suppose that T has a bounded everywhere defined inverse S with the property that for λ in the open unit disc of C the operator (I − λS) −1 exists and that the expression sup{(1 − |λ|) (I − λS) −1 : |λ| < 1} is finite. If T is power bounded, then so is S and hence in such a situation the operator T is similar to a unitary operator. If both the operators T * and S are square bounded in average, then again the operator T is similar to a unitary operator. Similar results hold for strongly 1991 Mathematics Subject Classification: 47A30, 47D05. Key words and phrases: power bounded operator, bounded semigroup, operator Poisson kernel, square bounded in average. The author is grateful to a number of people who in one way or another were involved during the preparation of this paper: F. Delbaen, L. Waelbroeck (Brussels), J. Zemánek (Warsaw). The author also wants to thank the National Fund for Scientific Research (NFWO) and the University of Antwerp (UIA) for their material support. The author is indebted to the referee for pointing out some errors in an earlier draft of the paper. Finally, the author is obliged to the "International Scientific and Technical Cooperation BLEU-Poland" for making it possible to visit the Banach Center in Warsaw in April 1994. The paper is in final form and no version of it will be published elsewhere. [59] 60 J. A. VAN CASTEREN continuous semigroups instead of (powers) of a single operator. Some results are also given in the more general Banach space context.

P-adic discrete semigroup of contractions

Proyecciones (Antofagasta), 2021

Let A ∈ B(X) be a spectral operator on a non-archimedean Banach space over Cp. In this paper, we give a necessary and sufficient condition on the resolvent of A so that the discrete semigroup consisting of powers of A is contractions.

Quasi-equicontinous exponential families of generalized function C-semigroups in locally convex spaces

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2018

Our main goal in this paper is to investigate the (q-)exponential C-distribution semigroups and (q-)exponential C-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We contribute to previous work and the work of many other authors, providing additionally plenty of various examples and applications of obtained results.

On quasi-contractivity of C0-semigroups on Banach spaces

Archiv der Mathematik, 2004

A basic result in semigroup theory states that every C 0-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula (e t n A P) n (where P denotes a bounded projection), we prove that whenever the generator A is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is not quasi-contractive. Mathematics subject classification (2000): 47A05, 47D06

On metrizable enveloping semigroups

Israel Journal of Mathematics, 2008

When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g ∈ G, in the compact space X X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily non-sensitive; (3) for any compatible metric d on X the metric d G (x, y) := sup{d(gx, gy) : g ∈ G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.