Second quantisation for skew convolution products of infinitely divisible measures

Suppose λ1 and λ2 are infinitely divisible Radon measures on real Banach spaces E1 and E2, respectively and let T : E1 → E2 be a Borel measurable mapping so that T(λ1) * ρ = λ2 for some Radon probability measure ρ on E2. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the "transition operator" PT : Lp(E2, λ2) → Lp(E1, λ1) given by [Formula: see text] as the second quantisation of a contraction operator acting between suitably chosen "reproducing kernel Hilbert spaces" associated with λ1 and λ2.

Isometries of a function space

It is proved here that an isometry on the subset of all positive functions ofL1⋂Lp(ℝ)can be characterized by means of a functionhtogether with a Borel measurable mappingϕofℝ, thus generalizing the Banach-Lamparti theorem ofLpspaces.