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Mat. Sb., 2001, Volume 192, Number 12, Pages 93–144 (Mi msb617)  

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 $\|A^jg\|\leqslant c(g)\xi^j$, $j=0,1,…$ . 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 $\gamma(\xi):=\min_{j=1,2,…(\xi\psi_j)^{1/j}$. Conditions for the sequence $\{\psi_j\}_{j=1}^\infty$ and the operator $A$ are found that ensure the equality
$$ \limsup_{j\to\infty}(\frac{\|A^jf\|}{\psi_j})^{1/j}=\limsup_{\xi\to\infty}\frac\xi{\gamma(E_\xi(f,A)^{-1})} . $$
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.

DOI: https://doi.org/10.4213/sm617

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English version:
Sbornik: Mathematics, 2001, 192:12, 1829–1876

Bibliographic databases:

UDC: 517.43+517.5
MSC: Primary 41A65; Secondary 46G20, 46B99, 47A05
Received: 06.02.2001

Citation: G. V. Radzievskii, “Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree”, Mat. Sb., 192:12 (2001), 93–144; Sb. Math., 192:12 (2001), 1829–1876

Citation in format AMSBIB
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\paper Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree
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\issue 12
\pages 93--144
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\pages 1829--1876
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