Functional calculus for a bounded C₀-semigroup on Hilbert space · arXivDesk2012.04440Dec 8, 2020This updated version is published in Journal of Functional Analysis
Functional calculus for a bounded C0-semigroup on Hilbert space
Loris Arnold, Christian Le Merdy
Abstract
We introduce a new Banach algebra A(C+) of bounded analytic functions on C+={z∈C:Re(z)>0}
which is an analytic version of the Figa-Talamenca-Herz algebras on
. Then we prove that the negative generator
of any bounded
-semigroup on Hilbert space
admits a bounded (natural) functional calculus
ρA:A(C+)→B(H) . We prove that this is an improvement of the bounded functional calculus
B(C+)→B(H) recently devised by Batty-Gomilko-Tomilov on a certain Besov algebra
B(C+) of analytic functions on
, by showing that
B(C+)⊂A(C+) and
B(C+)=A(C+) . In the Banach space setting, we give similar results for negative generators of
-bounded
-semigroups. The study of
A(C+) requires to deal with Fourier multipliers on the Hardy space
H1(R)⊂L1(R) of analytic functions.