Exponentially bounded C-semigroups and generation of semigroups

Abstract

This paper concerns exponentially bounded C-semigroups and semigroups of operators in a Banach space X. Recently Davies and Pang [l] introduced the notion of an exponentially bounded C-semigroup and characterized the generator of an exponentially bounded C-semigroup. On the other hand, Miyadera [7] discussed the complete infinitesimal generator of an exponentially bounded C-semigroup, and Tanaka [14] has characterized the complete infinitesimal generator of an exponentially bounded C-semigroup. There are many important works on semigroups of linear operators. (For example, see [4, 5, 12, 161.) Extending the notion of (C,)-semigroups, Oharu [9] introduced the semigroups of classes (Cc,,), k = 0, 1,2, . . . . and Da Prato [2] and Sobolevskii [13] introduced the semigroups of growth order c( 2 0. Generation theorems for semigroups of these classes have been studied in [2, 9, 10, 131. Our first purpose is to clarify the relationship between semigroups and exponentially bounded C-semigroups. It is shown that if {T(t); t 2 0} is a semigroup and C is an injective bounded linear operator with dense range satisfying r(t)C= CT(t) for t > 0 and R(C) c C= (xEX; lim r-o+ T(r)x = x}, then {CT(t); t 2 0) is an exponentially bounded C-semigroup whose complete infinitesimal generator is equal to that of {T(t); t 2 0). In particular, if { r(t); t k 0} is a semigroup of class (C,,,) or of growth order ~1, then {CT(t); t 2 O> becomes an exponentially bounded C-semigroup with a suitable C and its complete infinitesimal generator coincides with that of {T(t); t 2 O}. Our second and main purpose is to study the generation of semigroups via the theory of exponentially bounded C-semigroups. We shall characterize the complete infinitesimal generator of a semigroup {r(t); t 2 0}