On well-bounded operators of type (B)

Abstract

IntroductionThe notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.The main result of this paper (Theorem 5) is the characterisation of a type (B) operator by means of the weak compactness of its .s/j-operational calculus (as in Theorem 4.2 of (2)) and the description of the operational calculus using Stieltjes integrals of a kind suggested by Krabbe (5). Our results are also stronger than those of Berkson and Dowson in that we show thê -operational calculus for a type (B) operator to be continuous on pointwise convergent nets of uniformly bounded variation.We are indebted to H. R. Dowson for much valuable advice and encouragement during the preparation of this paper.

Spectral theory and operator ergodic theory on super-reflexive Banach spaces

“…The foregoing basic theory of spectral integration was developed in [38]. We refer the reader to §2 of [7] for a simplified account using the above notation.…”

“…Now, as P increases through the set of all partitions of [a, b] directed to increase by refinement, we have (see Lemma 4 of[38])…”

Spectral Theory of Linear Operators

“…The class of well-bounded operators on a Banach space was introduced by Smart were characterised by Berkson and Dowson [10] and by Spain [65] as being those for which the absolutely continuous functional calculus is weakly compact. Berkson and Dowson [10] introduced the class of well-bounded operators with decomposition of the identity of bounded variation.…”

“…be the family of projections on X whose adjoints are a decomposition of the identity of U, By lemmas 3.1.7 and 3.1.10, there exist a decomposition of the identity {F(s) :sE R} for Ul such that E(s)'F(s) = F(s)E(s)' for all sE lit Observe that U' and Ui have the single valued extension property (Lemma 1.3.3).The integrals described here are based on the modified Stieltjes integral of Krabbe[42]. Spain applied this integration theory to establish various characterisation of well-bounded operators of type (B) ([65], Theorem 5).…”

On some classes of AC-operators

“…This family of projections was called a decomposition of the identity (a definition will be given in section 2). Wellbounded operators of type (B) were characterised by Berkson and Dowson [2] and by Spain [15] as being those for which the absolutely continuous functional calculus is weakly compact. Berkson and Dowson [2] introduced the class of well-bounded operators with decomposition of the identity of bounded variation.…”