146 research outputs found
Product formulas in functional calculi for sectorial operators
We study the product formula in the framework of
(unbounded) functional calculus of sectorial operators . We give an abstract
result, and, as corollaries, we obtain new product formulas for the holomorphic
functional calculus, an extended Stieltjes functional calculus and an extended
Hille-Phillips functional calculus. Our results generalise previous work of
Hirsch, Martinez and Sanz, and Schilling.Comment: This is the authors accepted manuscript for a paper being published
in Mathematische Zeitschrift. The final publication is available at Springer
via http://dx.doi.org/10.1007/s00209-014-1378-
Fine scales of decay of operator semigroups
Motivated by potential applications to partial differential equations, we
develop a theory of fine scales of decay rates for operator semigroups. The
theory contains, unifies, and extends several notable results in the literature
on decay of operator semigroups and yields a number of new ones. Its core is a
new operator-theoretical method of deriving rates of decay combining
ingredients from functional calculus, and complex, real and harmonic analysis.
It also leads to several results of independent interest.Comment: Version 2 includes numerous minor corrections, and is the authors'
final version. The pape will be published in the Journal of the European
Mathematical Society in April 201
L^p-tauberian theorems and L^p-rates for energy decay
We prove -analogues of the classical tauberian theorem of Ingham and
Karamata, and its variations giving rates of decay. These results are applied
to derive -decay of operator families arising in the study of the decay of
energy for damped wave equations and local energy for wave equations in
exterior domains. By constructing some examples of critical behaviour we show
that the -rates of decay obtained in this way are best possible under our
assumptions.Comment: Minor corrections have been mad
A Besov algebra calculus for generators of operator semigroups and related norm-estimates
We construct a new bounded functional calculus for the generators of bounded
semigroups on Hilbert spaces and generators of bounded holomorphic semigroups
on Banach spaces. The calculus is a natural (and strict) extension of the
classical Hille-Phillips functional calculus, and it is compatible with the
other well-known functional calculi. It satisfies the standard properties of
functional calculi, provides a unified and direct approach to a number of
norm-estimates in the literature, and allows improvements of some of them.Comment: This is the authors' accepted version of a paper which will be
published in Mathematische Annale
Resolvent representations for functions of sectorial operators
We obtain integral representations for the resolvent of , where
is a holomorphic function mapping the right half-plane and the right
half-axis into themselves, and is a sectorial operator on a Banach space.
As a corollary, for a wide class of functions , we show that the operator
generates a sectorially bounded holomorphic -semigroup on a
Banach space whenever does, and the sectorial angle of is preserved.
When is a Bernstein function, this was recently proved by Gomilko and
Tomilov, but the proof here is more direct. Moreover, we prove that such a
permanence property for can be described, at least on Hilbert spaces, in
terms of the existence of a bounded -calculus for . As
byproducts of our approach, we also obtain new results on functions mapping
generators of bounded semigroups into generators of holomorphic semigroups and
on subordination for Ritt operators.Comment: The paper has been accepted for publication in Advances in
Mathematics. This is the authors' accepted versio
Optimal energy decay for the wave-heat system on a rectangular domain
We study the rate of energy decay for solutions of a coupled wave-heat system
on a rectangular domain. Using techniques from the theory of -semigroups,
and in particular a well-known result due to Borichev and Tomilov, we prove
that the energy of classical solutions decays like as .
This rate is moreover shown to be sharp. Our result implies in particular that
a general estimate in the literature, which predicts at least logarithmic decay
and is known to be best possible in general, is suboptimal in the special case
under consideration here. Our strategy of proof involves direct estimates based
on separation of variables and a refined version of the technique developed in
our earlier paper for a one-dimensional wave-heat system
Analytic Besov functional calculus for several commuting operators
This paper investigates analytic Besov functions of n variables which act on the generators of n commuting C0-semigroups on a Banach space. The theory for n = 1 has already been published, and the present paper uses a different approach to that case as well as extending to the cases when n ≥ 2. It also clarifies some spectral mapping properties and provides some operator norm estimates
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