Catalan generating functions for bounded operators

Springer Science and Business Media LLC - 2023
Pedro J. Miana1, Natalia Romero2
1Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain
2Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain

Tóm tắt

Abstract

In 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.

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Tài liệu tham khảo

Bartels, R.H., Stewart, G.W.: Algorithm 432: solution of the matrix equation $$AX + XB = C$$. Commun. Assoc. Comput. Mach. 15(9), 820–826 (1972)

Calvetti, D., Reichel, L.: Application of ADI iterative methods to the restoration of noisy images. SIAM J. Matrix Anal. Appl. 17, 165–186 (1996)

Chen, X., Chu, W.: Moments on Catalan numbers. J. Math. Anal. Appl. 349(2), 311–316 (2009)

Davis, G.J.: Numerical solution of a quadratic matrix equation. SIAM J. Sci. Stat. Comput. 2, 164–175 (1981)

Davis, G.J.: Algorithm 598: an algorithm to compute solvents of the matrix equation $$AX^2+BX+C=0$$. ACM Trans. Math. Softw. 9, 246–254 (1983)

Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)

Dungey, N.: Subordinated discrete semigroups of operators. Trans. Am. Math. Soc. 363(4), 1721–1741 (2011)

Gomilko, A., Tomilov, Y.: On discrete subordination of power bounded and Ritt operators. Indiana Univ. Math. J. 67(2), 781–829 (2018)

Gónzalez-Camus, J., Lizama, C., Miana, P.J.: Fundamental solutions for semidiscrete evolution equations via Banach algebras. Adv. Differ. Equ. 2021(35), 1–32 (2021)

Hernández-Verón, M.A., Romero, N.: Methods with prefixed order for approximating square roots with global and general convergence. Appl. Math. Comput. 194(2), 346–353 (2007)

Hernández-Verón, M.A., Romero, N.: Existence, localization and approximation of solution of symmetric algebraic Riccati equations. Comput. Math. Appl. 76(1), 187–203 (2018)

Kantorovich, L.V.: Functional Analysis and Applied Mathematics. Translated by C. D. Benster, National Bureau of Standards Report 1509 (1952)

Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications. Academic Press, Orlando (1985)

Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford Science Publications, Oxford (1995)

Larsen, R.: Banach Algebras: An Introduction. Marcel Dekker, New York (1973)

Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia (1999)

McFarland, J.E.: An iterative solution of the quadratic equation in Banach space. Proc. Am. Math. Soc. 9, 824–830 (1958)

Miana, P.J., Romero, N.: Moments of combinatorial and Catalan numbers. J. Number Theory 130(8), 1876–1887 (2010)

Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math. 3, 33–48 (1998)

Qi, F., Guo, B.-N.: Integral representations of the Catalan numbers and their applications. Mathematics 5(8), 1–31 (2017)

Rogers, L.C.G.: Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Probab. 4(2), 390–413 (1994)

Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, Inc., New York (1991)

Shapiro, L.W.: A Catalan triangle. Discret. Math. 14, 83–90 (1976)

Sloane, N.: A Handbook of Integer Sequences. Academic Press, New Jersey (1973)

Stanley, R.P.: Catalan Numbers. Cambridge University Press, Cambridge (2015)

Yosida, K.: Functional Analysis: Grundlehren der mathematischen Wissenchaften, vol. 123. Springer, Berlin (1980)

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