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

Let A be a linear operator in a complex Banach space X with domain D(A) and a non-empty resolvent set. An element g element of D∞(A):= intersectionj=0,1,...D(Aj) is called a vector of degree at most ζ(>0) with respect to A if ||Ajg||X≤c(g)ζj, j=0,1,.... The set of vectors of degree at most ζ is denoted by Gζ(A). The quantity Eζ(f,A)X=infgelementofGζ(A)||f-g||X is introduced and estimated in terms of the K-functional K(ζ-r,f;X,D(Ar))=infgelementofD(Ar)(||f-g||X+ζ-r||Arf||X) (the direct theorem). An estimate of this K-functional in terms of Eζ(f,A)X and ||f||x is established (the converse theorem). Using the estimates obtained, necessary and sufficient conditions for the following properties are found in terms of Eζ(f,A)X: 1) f element of D∞(A); 2) the series ezAf:=Σr=0∞(zrArf)/(r!) converges in some disc; 3) the series ezAf converges in the entire complex plane. The growth order and the type of the entire function ezAf are calculated in terms of Eζ(f,A)X