Joint BCAM-UPV/EHU Analysis and PDE seminar: Catalan operators in Operator Theory

Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$  is a power-bounded operator. The Catalan generating function is defined by the following Taylor series,

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

Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$  In this talk we study this algebraic equation in the case that $T$ is the infinitesimal generator of  a C_0-semigroup. We express $C(T)$ by means of an integral representations which involves the resolvent operator $(\lambda-T)^{-1}$ or the C_0-semigroup. In the case that $T$ is a bounded operator, we define  powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. Finally, we give some particular examples to illustrate our results and some ideas to continue this research in the future. This is a research proyect with Alejandro Mahillo (Universidad de Zaragoza) and Natalia Romero (Universidad de La Rioja).