Catalan generating functions for bounded operators

In this paper, we study the solution of the quadratic equation TY2 - 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 showthat a solution (named Catalan generating function) of the above equation is given by the Taylor series C(T) := Sigma(infinity)(n=0) CnT (n), where the sequence (Cn) (n >= 0) is thewell-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.