318 research outputs found
Regularity Properties of Sectorial Operators: Counterexamples and Open Problems
We give a survey on the different regularity properties of sectorial
operators on Banach spaces. We present the main results and open questions in
the theory and then concentrate on the known methods to construct various
counterexamples.Comment: 21 pages; Example 3.11 corrected; final versio
On continuity of solutions for parabolic control systems and input-to-state stability
We study minimal conditions under which mild solutions of linear evolutionary
control systems are continuous for arbitrary bounded input functions. This
question naturally appears when working with boundary controlled, linear
partial differential equations. Here, we focus on parabolic equations which
allow for operator-theoretic methods such as the holomorphic functional
calculus. Moreover, we investigate stronger conditions than continuity leading
to input-to-state stability with respect to Orlicz spaces. This also implies
that the notions of input-to-state stability and integral-input-to-state
stability coincide if additionally the uncontrolled equation is dissipative and
the input space is finite-dimensional.Comment: 19 pages, final version of preprint, Prop. 6 and Thm 7 have been
generalised to arbitrary Banach spaces, the assumption of boundedness of the
semigroup in Thm 10 could be droppe
The Weiss conjecture on admissibility of observation operators for contraction semigroups
We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functional C is infinite-time admissible if and only if there is an M > 0 such that parallel to IC(sI - A)(-1)parallel to less than or equal to M/root Re s for all s in the open right half-plane. Here A denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators
Mean ergodic theorems on norming dual pairs
We extend the classical mean ergodic theorem to the setting of norming dual
pairs. It turns out that, in general, not all equivalences from the Banach
space setting remain valid in our situation. However, for Markovian semigroups
on the norming dual pair (C_b(E), M(E)) all classical equivalences hold true
under an additional assumption which is slightly weaker than the e-property.Comment: 18 pages, 1 figur
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
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