  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 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 $$
C(T):=\sum_{n=0}^\infty C_nT^n, $$ 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.
