On the Infinite Product ofC0-Semigroups

✍ Scribed by W Arendt; A Driouich; O El-Mennaoui

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 334 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3341

✦ Synopsis

Given a family (e tAk ) t 0 (k # N) of commuting contraction semigroups, we investigate when the infinite product > k=1 e tAk converges and defines a C 0 -semigroup. A particular case is the heat semigroup in infinite dimension introduced by Cannarsa and Da Prato (J. Funct. Anal. 118 (1993), 22 42).

1998 Academic Press

1. Introduction

Recently, parabolic equations in infinite dimensions have received much attention in literature (see, for example, Pa Prato [DP] and Da Prato Zabczyk [DZ]). In particular, Cannarsa and Da Prato [CD1] showed that the Laplacian (with a certain weight) generates a semigroup on BUC(H), the space of all bounded uniformly continuous functions on a separable Hilbert space H, which is called the heat semigroup (see also [CD2]). This semigroup can be expressed as an infinite product,

of a commuting family of contraction semigroups (e tAk ) t 0 , k # N.

Motivated by this example, in the present paper, we start a systematic study of such infinite products. Already the simple example E=C, A k =i (k # N) shows that (1.1) does not always converge. In order to obtain positive results, one has to allow a ``change of speed'' represented by a article no.