Abstract
AbstractIn this paper, we study the solution of the quadratic equation$$TY^2-Y+I=0$$TY2-Y+I=0whereTis a linear and bounded operator on a Banach spaceX. We describe the spectrum set and the resolvent operator ofYin terms of the ones ofT. In the case that 4Tis a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series$$\begin{aligned} C(T):=\sum _{n=0}^\infty C_nT^n, \end{aligned}$$C(T):=∑n=0∞CnTn,where the sequence$$(C_n)_{n\ge 0}$$(Cn)n≥0is the well-known Catalan numbers sequence. We expressC(T) by means of an integral representation which involves the resolvent operator$$(\lambda T)^{-1}$$(λT)-1. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matricesTwhich involves Catalan numbers.
Publisher
Springer Science and Business Media LLC
Subject
Control and Optimization,Analysis,Algebra and Number Theory
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