Catalan Generating Functions for Generators of Uni-parametric Families of Operators

Abstract

AbstractIn this paper we study solutions of the quadratic equation $$AY^2-Y+I=0$$ A Y 2 - Y + I = 0 where A is the generator of a one parameter family of operator ($$C_0$$ C 0 -semigroup or cosine functions) on a Banach space X with growth bound $$w_0 \le \frac{1}{4}$$ w 0 ≤ 1 4 . In the case of $$C_0$$ 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, $$\begin{aligned} C(A)x := \int _{0}^\infty c(t) T(t)x \; \mathrm{d}t, \quad x\in X, \end{aligned}$$ C ( A ) x : = ∫ 0 ∞ c ( t ) T ( t ) x d t , x ∈ X , where c is the Catalan kernel, $$\begin{aligned} c(t) := \frac{1}{2\pi } \int _{\frac{1}{4}}^\infty e^{-\lambda t} \frac{\sqrt{4\lambda -1}}{\lambda } \; \mathrm{d}\lambda , \quad t>0. \end{aligned}$$ 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 $$L^1_{\omega }(\mathbb R^+)$$ L ω 1 ( R + ) , endowed with the usual convolution product, $$*$$ ∗ and with the cosine convolution product, $$*_c$$ ∗ c . The Hille–Phillips functional calculus allows to transfer these properties to $$C_0$$ 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.