Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.

Compact Hermitian operators on projective tensor products of Banach algebras

LetUandVbe, respectively, an infinite- and a finite-dimensional complex Banach algebras, and letU⊗pVbe their projective tensor product. We prove that (i) every compact Hermitian operatorT1onUgives rise to a compact Hermitian operatorTonU⊗pVhaving the properties that‖T1‖=‖T‖andsp(T1)=sp(T); (ii) ifUandVare separable andUhas Hermitian approximation property(HAP), thenU⊗pVis also separable and hasHAP; (iii) every compact analytic semigroup(CAS)onUinduces the existence of aCASonU⊗pVhaving some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.