Catalan generating functions for bounded operators

Abstract

In this paper, we study the solution of the quadratic equation TY^2-Y+I=0 where T is a linear and bounded operator on a Banach space X. We describe the spectrum set and the resolvent operator of Y in terms of the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series C(T):=∑n=0∞CnTn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} C(T):=\\sum _{n=0}^\\infty C_nT^n, \\end{aligned}$$\\end{document}where the sequence (C_n)_{nge 0} is the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator (lambda T)^{-1}. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan numbers.