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

• Published in: Mediterranean journal of mathematics 2022-10, Vol.19 (5), Article 238
• Main Authors: Mahillo, Alejandro, Miana, Pedro J.
• Format: Article
• Language: English
• Subjects: Banach spaces; Convolution; Functionals; Kernels; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Quadratic equations; Semigroups; Trigonometric functions
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-022-02155-7

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.