We show that every abelian Polish group is the topological factor group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.

Let $T=T\left(u\right):u\in {ℝ}_d^{+}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_1((\Omega ,\Sigma ,\mu );X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_1((\Omega ,\Sigma ,\mu );X)*=L_{\infty }((\Omega ,\Sigma ,\mu );X*)$, the adjoint semigroup $T*=T*\left(u\right):u\in {ℝ}_d^{+}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_{\infty }((\Omega ,\Sigma ,\mu );X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u\in {ℝ}_d^{+}$, has a contraction majorant P(u) defined on $L_1((\Omega ,\Sigma ,\mu );ℝ)$, that is, P(u) is a positive linear contraction on $L_1((\Omega ,\Sigma ,\mu );ℝ)$ such that $\Vert T\left(u\right)f\left(\omega \right)\Vert \le P\left(u\right)\Vert f\left(·\right)\Vert \left(\omega \right)$ almost everywhere...

The properties of a transformation $$f\mapsto {\tilde{f}}_h$$ by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that $$\left({\tilde{f}}_h\right)\widetilde{{}_k}={\tilde{f}}_{h+k}$$ for certain complex h and k, and that $$f\left(t\right)=\underset{h\to 0^{+}}{lim}{\tilde{f}}_h\left(t\right)$$ , where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous...

If A generates a bounded cosine function on a Banach space X then the negative square root B of A generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by B. The characterization relies on new results on the inversion of the vector-valued conjugate potential transform.

Let $(R{}_t$ be a transition semigroup of the Hilbert space-valued nonsymmetric Ornstein-Uhlenbeck process and let $\mu$ denote its Gaussian invariant measure. We show that the semigroup $(R{}_t$ is analytic in $L^2\left(\mu \right)$ if and only if its generator is variational. In particular, we show that the transition semigroup of a finite dimensional Ornstein-Uhlenbeck process is analytic if and only if the Wiener process is nondegenerate.

We are concerned with a relation between parabolicity and coerciveness in Besov spaces for a higher order linear evolution equation in a Banach space. As proved in a preceding work, a higher order linear evolution equation enjoys coerciveness in Besov spaces under a certain parabolicity condition adopted and studied by several authors. We show that for a higher order linear evolution equation coerciveness in Besov spaces forces the parabolicity of the equation. We thus conclude that parabolicity...

We prove results on ergodicity, i.e. on the property that the space is a direct sum of the kernel of an operator and the closure of its range, for closed linear operators A such that ${\left|\right|\alpha (\alpha -A)}^{-1}\left|\right|$ is uniformly bounded for all α > 0. We consider operators on Banach spaces which have the property that the space is complemented in its second dual space by a projection P. Results on ergodicity are obtained under a norm condition ||I - 2P|| ||I - Q|| < 2 where Q is a projection depending on the operator A....