Uniformly Γ ––radonifying Families of Operators and the Stochastic Weiss Conjecture

We introduce the notion of uniform γ –radonification of a family of operators, which unifies the notions of R–boundedness of a family of operators and γ –radonification of an individual operator. We study the properties of uniformly γ –radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem dU(t) = AU(t)dt +BdW (t), U(0) = 0. Here, A is the generator of a strongly continuous semigroup of operators on a Banach space E , B is a bounded linear operator from a separable Hilbert space H into E , and WH is an H – cylindrical Brownian motion. When A and B are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family { √ λR(λ ,A)B : λ ∈ Sθ} is uniformly γ –radonifying for some/all 0 < θ < 2 , where Sθ is the open sector of angle θ in the complex plane. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory. Mathematics subject classification (2010): Primary: 47B10, Secondary: 35R15, 47D06, 60H15, 93B28.