Norm Inequalities for Generators of Analytic Semigroups and Cosine Operator Functions

Abstract

We prove that if A is the infinitesimal generator of a bounded analytic semigroup in a sector {z 6 C : |argz| ^ (an)/2} of bounded linear operators on a Banach space, then the following inequalities hold: 114**11 S M<^<a)|| Jt || 1^* /">||A» J r||*/"(l S kû n-1)for any x G D(A n ) and for any 0 < (3 < a. This result helps us to answer in affirmative a question raised by M. W. Certain and T. G. Kurtz [3]. Similar inequalities are proved for cosine operator funtions.

A Landau–Kolmogorov inequality for generators of families of bounded operators

“…Moreover, the inequality (1.1) allows one to reduce the constant 4/3 to 6/5 in the inequality (1.2) when A generates a strongly continuous contraction cosine function, see [27,Theorem 3]. In fact, the proof of (1.1) is not elementary as commented in [26,Section 13] and [5, p. 227].…”

“…In Hilbert spaces, the optimal constant in (1.2) for a C 0 -contraction semigroup is 2 [10]; in C -Euclidean spaces it is treated in [13] and if A generates an analytic semigroup in [27]; see [10,25] and references therein for more details.…”

A survey of some norm inequalities

“…For additional results in the higher-order cases, see, for instance, [3], [9], [14], [42], [46], [64], [66], [72], [73, Ch. 1], [77], [81], [85], [86], [90], [91], [95].…”

“…[2, Sects. ) is treated in [90]. ⋄ Theorem 2.1 can be rewritten replacing the contraction semigroup by a uniformly bounded semigroup, that is, a semigroup such that for some M ≥ 1,…”

“…For additional results in the higher-order case, see, [3], [42], [60], [72], [73, Ch. 1], [77], [85], [90], [91], [92]. (iii) In the special case where X is a Hilbert space H in Theorem 3.1, the constant 2 on the right-hand side of (3.1) can be replaced by 1 (cf., [14]).…”