Powers of Catalan generating functions for bounded operators

Abstract

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 paper we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We obtain new formulae which involve Catalan triangle numbers; the spectrum of $c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ 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.

Figures (1)

• Figure 1: Sets $\partial (\sigma(c))$ in blue, $\partial (\sigma(b_1))$ in red and $\partial (\sigma(a_2))$ in green

Theorems & Definitions (29)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Remark 2.6