Catalan generating functions for bounded operators

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

$$\begin{aligned} C(T):=\sum _{n=0}^\infty C_nT^n, \end{aligned}$$

where the sequence \((C_n)_{n\ge 0}\) 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.