In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.
The famous Gowers tree space is the first example of a space not containing c₀, ℓ₁ or a reflexive subspace. We present a space with a similar construction and prove that it is hereditarily indecomposable (HI) and has ℓ₂ as a quotient space. Furthermore, we show that every bounded linear operator on it is of the form λI + W where W is a weakly compact (hence strictly singular) operator.
Let μ be a finite positive Borel measure on [0,1). Let be the Hankel matrix with entries . The matrix induces formally an operator on the space of all analytic functions in the unit disc by the fomula , z ∈ , where is an analytic function in . We characterize those positive Borel measures on [0,1) such that for all f in the Hardy space H¹, and among them we describe those for which is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².