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Catalan generating functions for bounded operators

Pedro J. Miana, Natalia Romero

Abstract

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 $(λ-T)^{-1}$. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices $T$ which involves Catalan numbers.

Catalan generating functions for bounded operators

Abstract

In this paper we study the solution of the quadratic equation where is a linear and bounded operator on a Banach space . We describe the spectrum set and the resolvent operator of in terms of operator . In the case that is a power-bounded operator, we show that a solution (named Catalan generating function) is given by the Taylor series where the sequence is the well-known Catalan numbers. We express by means of an integral representation which involves the resolvent operator . Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices which involves Catalan numbers.
Paper Structure (9 sections, 11 theorems, 76 equations, 2 figures, 1 table)

This paper contains 9 sections, 11 theorems, 76 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Some news results about Catalan numbers
  3. The sequence of Catalan numbers
  4. Inverse spectral mapping theorem of the quadratic Catalan equation
  5. Catalan generating functions for bounded operators
  6. Examples, applications and final comments
  7. Matrices on $\mathbb{C}^2$
  8. Catalan operators on $\ell^p$
  9. Iterative methods on $\mathbb{R}^n$ applied to Quasi-birth-death processes

Key Result

Theorem 2.1

Let $A$ be a commutative algebra over $\mathbb{R}$ or $\mathbb{C}$ with $x\in A$. If $y$ and $z$ are solutions of the quadratic equations then ${y+z\over 2}$ is a solution of the biquadratic equation $4x^2w^4-w^2+1=0$.

Figures (2)

  • Figure 1: The set $\partial(\sigma(c))$.
  • Figure 2: The set $\partial(\Omega)$ in blue and $\partial(\sigma(c))$ in red.

Theorems & Definitions (24)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • ...and 14 more