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Stochastic integration of functions with values in a Banach space

100%

van Neerven J., Weis L.

Studia Mathematica

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2005

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tom 166

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nr 2

131-170

EN

Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process ${W_H(t)}_{t∈[0,T]}$. The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution.

2

Operator theoretic properties of semigroups in terms of their generators

100%

Blunck S., Weis L.

Studia Mathematica

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2001

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tom 146

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nr 1

35-54

EN

Let $(T_{t})$ be a C₀ semigroup with generator A on a Banach space X and let 𝓐 be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to 𝓐 if and only if the integrated semigroup $S_{t}: = ∫_{0}^{t} T_{s}ds$ belongs to 𝓐. For analytic semigroups, $S_{t} ∈ 𝓐$ implies $T_{t} ∈ 𝓐$, and in this case we give precise estimates for the growth of the 𝓐-norm of $T_{t}$ (e.g. the trace of $T_{t}$) in terms of the resolvent growth and the imbedding D(A) ↪ X.

3

Norm continuity of $c_0$-semigroups

100%

Goersmeyer V., Weis L.

Studia Mathematica

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1999

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tom 134

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nr 2

169-178

EN

We show that a positive semigroup $T_t$ on $L_p(Ω,ν)$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of "smoothing properties" of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.

Ograniczanie wyników

3 Studia Mathematica

3 Weis L.

1 Blunck S.

1 Goersmeyer V.

1 van Neerven J.

1 2005

1 2001

1 1999

Stochastic integration of functions with values in a Banach space

Operator theoretic properties of semigroups in terms of their generators

Norm continuity of $c_0$-semigroups