| Abstrakt: |
In this paper we study solutions of the quadratic equation A Y 2 - Y + I = 0 where A is the generator of a one parameter family of operator ( C 0 -semigroup or cosine functions) on a Banach space X with growth bound w 0 ≤ 1 4 . In the case of C 0 -semigroups, we show that a solution, which we call Catalan generating function of A, C(A), is given by the following Bochner integral, C (A) x : = ∫ 0 ∞ c (t) T (t) x d t , x ∈ X , where c is the Catalan kernel, c (t) : = 1 2 π ∫ 1 4 ∞ e - λ t 4 λ - 1 λ d λ , t > 0. Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernel c as an element in Banach algebras L ω 1 (R +) , endowed with the usual convolution product, ∗ and with the cosine convolution product, ∗ c . The Hille–Phillips functional calculus allows to transfer these properties to C 0 -semigroups and cosine functions. In particular, we obtain a spectral mapping theorem for C(A). Finally, we present some examples, applications and conjectures to illustrate our results. [ABSTRACT FROM AUTHOR] |
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