On extensions and restrictions of semigroups of linear operators in a Banach space and their applications

Abstract

Key words Differential equation in a Banach space, classical solution, weak solution, locally convex space, C0 -semigroup, extension and restriction of a C0 -semigroup, exponential operator-valued function, locally analytic vector-valued function, Cauchy problem, Cauchy-Kovalevskaya theorem MSC (2010) 34G10, 47D06 Dedicated to Eduard Tsekanovskiȋ in honour of his 75th birthdayThe description of all solutions on (0, ∞) of differential equations of parabolic and elliptic type in a Banach space is given without imposing any conditions on their behavior near the point 0, and the properties of them are studied. The necessary and sufficient conditions on the initial data, under which the Cauchy problem for a general operator-differential equation is solvable in the class of locally analytic, entire or entire of finite order and finite type vector-valued functions, are established. The results are based on the theory of restrictions and extensions of analytic semigroups set forth below.y Br (B) = max z ∈Or y(z) .