Abstract
The paper provides new characterisations of generators of cosine functions and C 0-groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0-groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini’s result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh’s characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0-semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.
Highlights
Recall that for a Banach space X, a strongly continuous function C : R → L(X ) is called a cosine function if it satisfies d’Alembert’s equationC(t + s) + C(t − s) = 2C(t)C(s), t, s ∈ R, and C(0) = I
An operator A on X is the generator of a cosine function if and only if (ω, ∞) ⊂ ρ(A) for some ω ∈ R and (·)R((·)2, A) is the Laplace transform of an operator-valued, strongly continuous function on R+
In the first part of this paper, we provide characterisations of generators of cosine functions and C0-groups on UMD spaces which, in particular, throw a new light on the phenomenon of the group decomposition of cosine functions, see Theorems
Summary
Recall that for a Banach space X , a strongly continuous function C : R → L(X ) is called a cosine function if it satisfies d’Alembert’s equation. The following characterisations of generators of cosine functions and C0-groups on UMD spaces are the main results of this section. Recall that the complex inversion formula for C0-semigroups holds on every UMD space, see [14] or [2, Section 3.12] It shows that there exists a generator A of a C0-(semi)group such that for any σ > 0 and p > 1 the following condition. 4.3] Fattorini stated without proof that a weaker asymptotic behaviour of the resolvent of a densely defined linear operator A on a Hilbert space X , weaker than the one from (ii) of Theorem 2.4, is sufficient to ensure that A generates a cosine function. This sheds some light on the role of the assumptions on the resolvent of the adjoint of A in Theorem 2.8 and [27, Theorem 4.1]
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