Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree

Let $A$ be a linear operator with domain $\mathfrak D(A)$ in a complex Banach space $X$. An element $g\in\mathfrak D_\infty(A):=\bigcap_{j=0}^\infty\mathfrak D(A^j)$ is called a vector of degree at most $\xi$ $(>0)$ relative to $A$ if $\|A^jg\|\leqslant c(g)\xi^j$, $j=0,1,\dots$ . The set of vectors of degree at most $\xi$ is denoted by $\mathfrak G_\xi(A)$ and the least deviation of an element $f$ of $X$ from the set $\mathfrak G_\xi(A)$ is denoted by $E_\xi(f,A)$. For a fixed sequence of positive numbers $\{\psi_j\}_{j=1}^\infty$ consider a function $\gamma(\xi):=\min_{j=1,2,\dots}(\xi\psi_j)^{1/j}$. Conditions for the sequence $\{\psi_j\}_{j=1}^\infty$ and the operator $A$ are found that ensure the equality
$$ \limsup_{j\to\infty}\biggl(\frac{\|A^jf\|}{\psi_j}\biggr)^{1/j}=\limsup_{\xi\to\infty}\frac\xi{\gamma(E_\xi(f,A)^{-1})}\,. $$
for $f\in\mathfrak D_\infty(A)$. If the quantity on the left-hand side of this formula is finite, then $f$ belongs to the Hadamard class determined by the operator $A$ and the sequence $\{\psi_j\}_{j=1}^\infty$. One consequence of the above formula is an expression in terms of $E_\xi(f,A)$ for the radius of holomorphy of the vector-valued function $F(zA)f$, where $f\in\mathfrak D_\infty(A)$, and $F(z):=\sum_{j=1}^\infty z^j/\psi_j$ is an entire function.