000127559 001__ 127559 000127559 005__ 20241125101142.0 000127559 0247_ $$2doi$$a10.1007/s43034-023-00290-0 000127559 0248_ $$2sideral$$a134550 000127559 037__ $$aART-2023-134550 000127559 041__ $$aeng 000127559 100__ $$0(orcid)0000-0001-9430-343X$$aMiana, Pedro J.$$uUniversidad de Zaragoza 000127559 245__ $$aCatalan generating functions for bounded operators 000127559 260__ $$c2023 000127559 5060_ $$aAccess copy available to the general public$$fUnrestricted 000127559 5203_ $$aIn this paper, we study the solution of the quadratic equation TY2−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 the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series C(T):=∑n=0∞CnTn, where the sequence (Cn)nis the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator (λT)−1. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan numbers 000127559 540__ $$9info:eu-repo/semantics/openAccess$$aby$$uhttp://creativecommons.org/licenses/by/3.0/es/ 000127559 590__ $$a1.2$$b2023 000127559 592__ $$a0.523$$b2023 000127559 591__ $$aMATHEMATICS$$b80 / 490 = 0.163$$c2023$$dQ1$$eT1 000127559 593__ $$aAlgebra and Number Theory$$c2023$$dQ2 000127559 591__ $$aMATHEMATICS, APPLIED$$b140 / 332 = 0.422$$c2023$$dQ2$$eT2 000127559 593__ $$aControl and Optimization$$c2023$$dQ2 000127559 593__ $$aAnalysis$$c2023$$dQ2 000127559 594__ $$a2.0$$b2023 000127559 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion 000127559 700__ $$aRomero, Natalia 000127559 7102_ $$12006$$2015$$aUniversidad de Zaragoza$$bDpto. Matemáticas$$cÁrea Análisis Matemático 000127559 773__ $$g14, 69 (2023), 21$$pAnn. funct. analysis$$tAnnals of functional analysis$$x2639-7390 000127559 8564_ $$s733190$$uhttps://zaguan.unizar.es/record/127559/files/texto_completo.pdf$$yVersión publicada 000127559 8564_ $$s963809$$uhttps://zaguan.unizar.es/record/127559/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada 000127559 909CO $$ooai:zaguan.unizar.es:127559$$particulos$$pdriver 000127559 951__ $$a2024-11-22-12:03:00 000127559 980__ $$aARTICLE