Powers of Catalan generating functions for bounded operators

Abstract

Let c=false(Cnfalse)n≥0$$ c={\left({C}_n\right)}_{n\ge 0} $$ be the Catalan sequence and T$$ T $$ a linear and bounded operator on a Banach space X$$ X $$ such 4T$$ 4T $$ is a power‐bounded operator. The Catalan generating function is defined by the following Taylor series: Cfalse(Tfalse):=∑n=0∞CnTn.$$ C(T):= \sum \limits_{n=0}^{\infty }{C}_n{T}^n. $$ Note that the operator Cfalse(Tfalse)$$ C(T) $$ is a solution of the quadratic equation TY2−Y+I=0$$ T{Y}^2-Y+I=0 $$. In this paper, we define powers of the Catalan generating function Cfalse(Tfalse)$$ C(T) $$ in terms of the Catalan triangle numbers. We obtain new formulae that involve Catalan triangle numbers: the spectrum of c∗j$$ {c}^{\ast j} $$ and the expression of c−∗j$$ {c}^{-\ast j} $$ for j≥1$$ j\ge 1 $$ in terms of Catalan polynomials ( ∗$$ \ast $$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.