153 research outputs found
Counterexamples to the discrete and continuous weighted Weiss conjectures
Counterexamples are presented to weighted forms of the Weiss conjecture in
discrete and continuous time. In particular, for certain ranges of ,
operators are constructed that satisfy a given resolvent estimate, but fail to
be -admissible. For the operators constructed are
normal, while for the operator is the unilateral shift on
the Hardy space .Comment: 16 page
β-admissibility of observation operators for hypercontractive semigroups
We prove a Weiss conjecture on β -admissibility of observation operators for discrete and continuous γ -hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case γ=2 , which corresponds to the unweighted Bergman shift
The stochastic Weiss conjecture for bounded analytic semigroups
Suppose -A admits a bounded H-infinity calculus of angle less than pi/2 on a
Banach space E with Pisier's property (alpha), let B be a bounded linear
operator from a Hilbert space H into the extrapolation space E_{-1} of E with
respect to A, and let W_H denote an H-cylindrical Brownian motion. Let
gamma(H,E) denote the space of all gamma-radonifying operators from H to E. We
prove that the following assertions are equivalent:
(i) the stochastic Cauchy problem dU(t) = AU(t)dt + BdW_H(t) admits an
invariant measure on E;
(ii) (-A)^{-1/2} B belongs to gamma(H,E);
(iii) the Gaussian sum \sum_{n\in\mathbb{Z}} \gamma_n 2^{n/2} R(2^n,A)B
converges in gamma(H,E) in probability.
This solves the stochastic Weiss conjecture proposed recently by the second
and third named authors.Comment: 17 pages; submitted for publicatio
- …