Direct and converse theorems in problems of approximation by vectors of finite degree

Abstract

Let be a linear operator in a complex Banach space with domain and a non-empty resolvent set. An element is called a vector of degree at most with respect to if , . The set of vectors of degree at most is denoted by . The quantity is introduced and estimated in terms of the -functional (the direct theorem). An estimate of this -functional in terms of and is established (the converse theorem). Using the estimates obtained, necessary and sufficient conditions for the following properties are found in terms of : 1) ; 2) the series converges in some disc; 3) the series converges in the entire complex plane. The growth order and the type of the entire function are calculated in terms of .