Dissipative operators and series inequalities

Herbert A. Gindler, Jerome A. Goldstein
1981 Bulletin of the Australian Mathematical Society  
Of concern is the best constant K in the inequality \\Ax\\ 2 K\\A a;||||a;|| where A generates a strongly continuous contraction semigroup in a Hirbert space. Criteria are obtained for approximate extremal vectors x when K = 2 ( J 5 2 always holds). By specializing A + I to be a shift operator on a sequence space, very simple proofs of Copson's recent results on series inequalities follow. Inequalities of the above type are also studied on IF spaces, and earlier results of the authors and of Ho
more » ... I brook are improved. tA i whence the semigroup [e : t 2 0} generated by A is contractive, and so A is m-dissipative [JO]. The inequality C(X; A) 5 1* for m-dissipative operators A was proved by Kail man and Rota [5] . Kato [6] showed that C(X; A) < 2 holds if X is a Hilbert space. If A (or L ) i s normal, then ||An|| 2 = (Ax, Ax> = <A*Ax, x) < ||i4*Ac||||x|| = ||i4 2 x||||x|| because \\A*y\\ = \\Ay\\ by normality. Thus C(X; A) < 1 . This was noted available at https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0004972700007309 fatcat:zio54bvd6jeojddxzubfs7syoe