Page 193 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 193
OPERATOR SEMIGROUPS AND EVOLUTION EQUATIONS (MS-29)
Fractional semidiscrete evolution equations in Lebesgue sequence spaces
Pedro J. Miana, pjmiana@unizar.es
Universidad de Zaragoza, Spain
In this talk, we give representations for solutions of time-fractional differential equations that
involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving
kernels through the discrete Fourier transform. We consider finite difference operators of first
and second orders, which are generators of uniformly continuous semigroups and cosine func-
tions. We present the linear and algebraic structures (in particular, factorization properties) and
their norms and spectra in the Lebesgue space of summable sequences. We identify fractional
powers of these generators and apply to them the subordination principle. We also give some
applications and consequences of our results. These results have been published in a joint paper
with Carlos Lizama and Jorge González-Camus from the Universidad de Santiago de Chile.
Evolution equations on graph and networks: diffusion and beyond
Delio Mugnolo, deliomu@gmail.com
FernUniversität in Hagen, Germany
I will present several models of hyperbolic and parabolic systems supported on metric graphs.
Well-posedness and long-time behavior can be studied by an interplay of variational, functional
analytical, and spectral theoretical methods. If time allows, I will also present a new class of
functionals and use it to describe the diffusive efficiency of a given graph topology.
Weighted composition operators via hyperbolic C0-groups on D
Jesús Oliva Maza, joliva@unizar.es
Universidad de Zaragoza, Spain
Coauthors: Luciano Abadías, José E. Galé, Pedro J. Miana
In this talk, we present our recent work regarding a family of C0-groups of weighted composi-
k◦ϕt
tion operators on Hardy and Bergman spaces on the disk, i.e. f → k f ◦ ϕt, where (ϕt)t∈R is
a group of hyperbolic automorphisms of the disk D onto itself, and k : D → C is a holomorphic
function with polynomial limits of arbitrary order α, β ∈ R at the fixed points of (ϕt)t∈R.
In the first part, we are able to characterize in detail the spectra of the infinitesimal generator
of the C0-group, which only depends on the values of α and β. More precisely, we obtain its
point spectra, approximate point spectra and essential spectra. This is done by using a mixture
of tools of C0-semigroups theory, and the theory of weighted composition operators on these
spaces of holomorphic functions.
After that, an application of the spectral mapping theorem is shown with some subordinated
operators, some of which resemble the Hilbert or Cesàro operators. In a final remark, we
show that one cannot take these C0-groups to p(N0) spaces since they do not define bounded
operators there for p = 2.
191
Fractional semidiscrete evolution equations in Lebesgue sequence spaces
Pedro J. Miana, pjmiana@unizar.es
Universidad de Zaragoza, Spain
In this talk, we give representations for solutions of time-fractional differential equations that
involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving
kernels through the discrete Fourier transform. We consider finite difference operators of first
and second orders, which are generators of uniformly continuous semigroups and cosine func-
tions. We present the linear and algebraic structures (in particular, factorization properties) and
their norms and spectra in the Lebesgue space of summable sequences. We identify fractional
powers of these generators and apply to them the subordination principle. We also give some
applications and consequences of our results. These results have been published in a joint paper
with Carlos Lizama and Jorge González-Camus from the Universidad de Santiago de Chile.
Evolution equations on graph and networks: diffusion and beyond
Delio Mugnolo, deliomu@gmail.com
FernUniversität in Hagen, Germany
I will present several models of hyperbolic and parabolic systems supported on metric graphs.
Well-posedness and long-time behavior can be studied by an interplay of variational, functional
analytical, and spectral theoretical methods. If time allows, I will also present a new class of
functionals and use it to describe the diffusive efficiency of a given graph topology.
Weighted composition operators via hyperbolic C0-groups on D
Jesús Oliva Maza, joliva@unizar.es
Universidad de Zaragoza, Spain
Coauthors: Luciano Abadías, José E. Galé, Pedro J. Miana
In this talk, we present our recent work regarding a family of C0-groups of weighted composi-
k◦ϕt
tion operators on Hardy and Bergman spaces on the disk, i.e. f → k f ◦ ϕt, where (ϕt)t∈R is
a group of hyperbolic automorphisms of the disk D onto itself, and k : D → C is a holomorphic
function with polynomial limits of arbitrary order α, β ∈ R at the fixed points of (ϕt)t∈R.
In the first part, we are able to characterize in detail the spectra of the infinitesimal generator
of the C0-group, which only depends on the values of α and β. More precisely, we obtain its
point spectra, approximate point spectra and essential spectra. This is done by using a mixture
of tools of C0-semigroups theory, and the theory of weighted composition operators on these
spaces of holomorphic functions.
After that, an application of the spectral mapping theorem is shown with some subordinated
operators, some of which resemble the Hilbert or Cesàro operators. In a final remark, we
show that one cannot take these C0-groups to p(N0) spaces since they do not define bounded
operators there for p = 2.
191
