119916 20240319081017.0 doi 10.1007/s00009-022-02155-7 sideral 130677 ART-2022-130677 eng Mahillo, Alejandro Universidad de Zaragoza (orcid)0000-0003-4189-0268 Catalan Generating Functions for Generators of Uni-parametric Families of Operators 2022 Access copy available to the general public Unrestricted In this paper we study solutions of the quadratic equation AY2−Y+I=0 where A is the generator of a one parameter family of operator (C0-semigroup or cosine functions) on a Banach space X with growth bound w0≤14. In the case of C0-semigroups, we show that a solution, which we call Catalan generating function of A, C(A), is given by the following Bochner integral, C(A)x:=∫∞0c(t)T(t)xdt,x∈X, where c is the Catalan kernel, c(t):=12π∫∞14e−λt4λ−1−−−−−√λdλ,t>0. Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernel c as an element in Banach algebras L1ω(R+), endowed with the usual convolution product, ∗ and with the cosine convolution product, ∗c. The Hille–Phillips functional calculus allows to transfer these properties to C0-semigroups and cosine functions. In particular, we obtain a spectral mapping theorem for C(A). Finally, we present some examples, applications and conjectures to illustrate our results. info:eu-repo/grantAgreement/ES/DGA/E48-20R info:eu-repo/grantAgreement/ES/MCYTS-DGI-FEDER/PID2019-105979GB-I00 info:eu-repo/semantics/openAccess by http://creativecommons.org/licenses/by/3.0/es/ 1.1 2022 MATHEMATICS 101 / 329 = 0.307 2022 Q2 T1 MATHEMATICS, APPLIED 161 / 267 = 0.603 2022 Q3 T2 0.531 2022 Mathematics (miscellaneous) 2022 Q2 1.7 2022 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Miana, Pedro J. Universidad de Zaragoza (orcid)0000-0001-9430-343X 2006 015 Universidad de Zaragoza Dpto. Matemáticas Área Análisis Matemático 19, 5 (2022), 238 [27 pp.] Mediterranean Journal of Mathematics Mediterranean Journal of Mathematics 1660-5446 651847 http://zaguan.unizar.es/record/119916/files/texto_completo.pdf Versión publicada 1006291 http://zaguan.unizar.es/record/119916/files/texto_completo.jpg?subformat=icon icon Versión publicada oai:zaguan.unizar.es:119916 articulos driver 2024-03-18-15:45:05 ARTICLE