SEMIGROUPS CALCULUS · Functional Calculus for operators and semigroups under resolvent or boundedness conditions
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2004-07-01 → 2005-06-30
- EU contribution
- €150,826
- Participants
- 1
- Scheme
- EIF
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Results in brief
Final Activity Report Summary - SEMIGROUPS CALCULUS (Functional Calculus for operators and semigroups under resolvent or boundedness conditions)
We studied the functional calculus for bounded holomorphic semigroups (i.e. for sectorial operators) and their discrete analogue, Tadmor-Ritt operators. This subject is closely linked with the question of maximal regularity of PDE's associated to the semigroup. It is well-known that, in general, no $H^\infty$ calculus exists, $H^\infty$ being the algebra of all bounded holomorphic functions on the right half plane. We proved that a $B^0_{\infty 1}$ calculus exists, $B^0_{\infty 1}$ being the non-homogeneous Besov algebra, that is, the subalgebra of $H^\infty$ consisting of functions $f$ satisfying $\int_0^\infty \max_{y \in \mathbb{R}} |f'(x+iy)| dx < \infty$, endowed with the corresponding norm. The same holds for Tadmor-Ritt operators with the usual Besov algebra $B^0_{\infty 1}$ on the unit disc.
Data: CORDIS, © European Union
Project objective
It is planned to obtain some estimates of a functional calculus (defined in terms of the Lap lace transform) for semi groups. To this aim the duality method successfully used by the applicant for the single operator case (with the spectrum in the unit disc) shall be applied. Most interesting is the case of bounded semi groups and semi groups whose generators (with the spectrum in the left half plane) satisfy some resolving growth conditions (analogues of Kris and Tadmor-Ritt conditions for a single operator). In particular, in the case of entire functions of a given exponential type (which are analogues of the polynomials of a given degree), one can hope to obtain some estimates in terms of the type. Some intermediate stages will consist in finding a half plane version of the notions and theorems that have been used in the case of a single operator: the Besot classes and the spaces of Cauchy-Stieltjes integrals and their multipliers, the Rises turndown collar theorem on the uniform convergence of power series, etc. It is intended to study the sharpness of the estimates obtained in the case of semi groups, and also in the single operator case where the question remains unanswered (i.e., for operators on Hilbert spaces). This, probably, could be done with the help of the functional model and Foguel-Hankel operators. Applications to evolution equations shall be elaborated.
Original text from CORDIS.
Participants
- UNIVERSITAET ULM · ULMCoordinatorGermany
Links
Data: CORDIS, © European Union