Catalan generating functions for bounded operators

Abstract

In this paper we study the solution of the quadratic equation $T{Y}^2-Y+I=0$ 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 operator $T$. In the case that $4T$ is a power-bounded operator, we show that a solution (named Catalan generating function) is given by the Taylor series $$C(T):=\sum _{n=0}^{\infty }{C}_n{T}^n,$$ where the sequence $(C_n{)}_{n\ge 0}$ is the well-known Catalan numbers. 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.

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