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

Abstract

In this paper we study solutions of the quadratic equation AY^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 le frac{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)xdt,x∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} C(A)x := \\int _{0}^\\infty c(t) T(t)x \\; \\mathrm{d}t, \\quad x\\in X, \\end{aligned}$$\\end{document}where c is the Catalan kernel, c(t):=12π∫14∞e-λt4λ-1λdλ,t>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} c(t) := \\frac{1}{2\\pi } \\int _{\\frac{1}{4}}^\\infty e^{-\\lambda t} \\frac{\\sqrt{4\\lambda -1}}{\\lambda } \\; \\mathrm{d}\\lambda , \\quad t>0. \\end{aligned}$$\\end{document}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^+), 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.