Generators of C0-Semigroups of weighted composition operators

We prove that in a large class of Banach spaces of analytic functions inthe unit disc D an (unbounded) operator Af = G · f′ + g · f with G, g analytic in D generates a C0-semigroup of weighted composition operators if and only if it generates a C0-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces X of analytic functions in D contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a C0-semigroup of weighted composition operators on X. This generalizes previous results in [6] and [1] regarding C0-semigroup of (unweighted) composition operators

Generators of C0-semigroups of weighted composition operators

We prove that in a large class of Banach spaces of analytic functions in the unit disc ⅅ an (unbounded) operator Af = G · f′ + g · f with G, g analytic in ⅅ generates a C0-semigroup of weighted composition operators if and only if it generates a C0-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces X of analytic functions in ⅅ contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a C0-semigroup of weighted composition operators on X. This generalizes previous results in [7] and [1] regarding C0-semigroup of (unweighted) composition operator

Generic non-extendability and total unboundedness in function spaces

For a function space X(Ω) satisfying weak assumptions we prove that the generic function in X(Ω) is totally unbounded, hence non-extendable. We provide several examples of such spaces; they are mainly localized versions of classical function spaces and intersections of them. © 2019 Elsevier Inc