A Desch–Schappacher perturbation theorem for bi‐continuous semigroups

Abstract

We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra ℒ( ) of bounded, linear operators on a Banach space , and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem. K E Y W O R D Sbi-continuous semigroups, Desch-Schappacher type perturbation, extrapolation spaces, implemented semigroups M S C ( 2 0 1 0 ) 34G10, 46A70, 47A55, 47D03 INTRODUCTIONAs suggested by Greiner in [17] abstract perturbation theory of one-parameter semigroups provides good means to change the domain of a semigroup generator. For this an enlargement of the underlying Banach space may be necessary and extrapolation spaces become important. One of the well-known results in this direction goes back to the papers of Desch and Schappacher, see [7] and [8]. Another prominent example of such general perturbation techniques is due to Staffans and Weiss, [30,31], and an elegant abstract operator theoretic/algebraic approach has been developed by Adler, Bombieri and Engel in [1]. A general theory of unbounded domain perturbations is given by Hadd, Manzo and Rhandi [18]. A more recent paper by Bátkai, Jacob, Voigt and Wintermayr [4] extends the notion of positivity to extrapolation spaces, and studies positive perturbations for positive semigroups on AM-spaces. Hence, the study of abstract Desch-Schappacher type perturbations is a lively research field, to which we contribute with the present article. The reason for such an active interest in this area is that the range of application is vast. We mention here only a selection from the most recent ones: boundary perturbations by Nickel [29], boundary feedback by Casarino, Engel, Nagel and Nickel [6], boundary control by Engel, Kramar Fijavž, Klöss, Nagel and Sikolya [11] and Engel and Kramar Fijavž [10], port-Hamiltonian systems by Baroun and Jacob [3], control theory by Jacob, Nabiullin, Partington and Schwenninger [19,20] and Jacob, Schwenninger and Zwart [21] and vertex control in networks by Engel and Kramar Fijavž [9,12].All the previously mentioned abstract perturbation results were developed for strongly continuous semigroups of linear operators on Banach spaces, 0 -semigroups for short. This is, for certain applications, e.g., for the theory of Markov transition semigroups (see [27], for a more general theory we refer to [26]), far too restrictive. For this situation the Banach space of

A Lumer-Phillips type generation theorem for bi-continuous semigroups

“…Since its outset in 2001, the theory of bi-continuous semigroups has grown and by now includes a Hille-Yosida type generation theorem, an approximation theory as well as a perturbation theory including inter-and extrapolation spaces [3,5,6,7,13,15,16]. In this paper we add another piece to the puzzle by establishing a Lumer-Phillips type generation theorem for contraction semigroups.…”

Semigroups for flows on limits of graphs

“…We mentioned in the introduction that the theory of flows in networks has been generalized by M. Kramar Fijavž and the author in [14] to a bigger class of operator semigroups on the phase space L ∞ [0, 1] , 1 , known as bi-continuous semigroups. These objects have a rich structure and have been introduced by F. Kühnemund [21] and further developed by B. Farkas and the author [11][12][13]. We will not go into the details of this theory since this is not the topic of this paper.…”

Semigroups for flows on limits of graphs

“…We mentioned in the introduction, that the theory for flows in networks has been generalized by M. Kramar Fijavž and the author in [15] to a bigger class of operator semigroups on the phase space L ∞ [0, 1] , ℓ 1 , the so-called bi-continuous semigroups. These objects have a rich structure and have been introduced by F. Kühnemund [22] and further developed by B. Farkas and the author [13,14,12]. We will not go into the details of this theory since this is not the topic of this paper.…”