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 $\mathfrak D(A)$ and a non-empty resolvent set. An element $g\in \mathfrak D_\infty (A):=\bigcap _{j=0,1,\dots }\mathfrak D(A^j)$ is called a vector of degree at most $\zeta (>0)$ with respect to $A$ if $\|A^jg\|_X\leqslant c(g)\zeta ^j$, $j=0,1,\dots $ . The set of vectors of degree at most $\zeta$ is denoted by $\mathfrak G_\zeta (A)$. The quantity $E_\zeta (f,A)_X=\inf _{g\in \mathfrak G_\zeta (A)}\|f-g\|_X$ is introduced and estimated in terms of the $K$-functional $K\bigl (\zeta ^{-r},f;X,\mathfrak D(A^r)\bigr ) =\inf _{g\in \mathfrak D(A^r)}\bigl (\|f-g\|_X+\zeta ^{-r}\|A^rf\|_X\bigr )$ (the direct theorem). An estimate of this $K$-functional in terms of $E_\zeta (f,A)_X$ 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 $E_\zeta (f,A)_X$: 1) $f\in \mathfrak D_\infty (A)$; 2) the series $e^{zA}f:=\sum _{r=0}^\infty (z^rA^rf)/(r!)$ converges in some disc; 3) the series $e^{zA}f$ converges in the entire complex plane. The growth order and the type of the entire function $e^{zA}f$ are calculated in terms of $E_\zeta (f,A)_X$.