Catalan Generating Functions for Generators of Uni-parametric Families of Operators

Mahillo, Alejandro (Universidad de Zaragoza) ; Miana, Pedro J. (Universidad de Zaragoza)

Resumen: 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.

Idioma: Inglés

DOI: 10.1007/s00009-022-02155-7

Año: 2022

Publicado en: Mediterranean Journal of Mathematics 19, 5 (2022), 238 [27 pp.]
ISSN: 1660-5446

Factor impacto JCR: 1.1 (2022)

Categ. JCR: MATHEMATICS rank: 101 / 329 = 0.307 (2022) - Q2 - T1
Categ. JCR: MATHEMATICS, APPLIED rank: 161 / 267 = 0.603 (2022) - Q3 - T2

Factor impacto CITESCORE: 1.7 - Mathematics (Q3)

Factor impacto SCIMAGO: 0.531 - Mathematics (miscellaneous) (Q2)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-20R
Financiación: info:eu-repo/grantAgreement/ES/MCYTS-DGI-FEDER/PID2019-105979GB-I00

Tipo y forma: Article (Published version)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)