Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree
G. V. Radzievskii Institute of Mathematics, Ukrainian National Academy of Sciences
Abstract:
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.
UDC:
517.43+517.5
MSC: Primary
41A65; Secondary
46G20,
46B99,
47A05 Received: 06.02.2001
DOI:
10.4213/sm617