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

Journal Article · Mon Dec 31 04:00:00 EST 2001 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2001V192N12ABEH000617; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21205651

Radzievski, G V ^[1]

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1. Institute of Mathematics of Ukrainian National Academy of Sciences, Kiev (Ukraine)

Let A be a linear operator with domain D(A) in a complex Banach space X. An element g element of D{sub {infinity}}(A):=intersection{sub j=1}{sup {infinity}}D(A{sup j}) is called a vector of degree at most {xi} (>0) relative to A if ||A{sup j}g||{<=}c(g){xi}{sup j}, j=0,1,.... The set of vectors of degree at most {xi} is denoted by G{sub {xi}}(A) and the least deviation of an element f of X from the set G{sub {xi}}(A) is denoted by E{sub {xi}}(f,A). For a fixed sequence of positive numbers ({psi}{sub j}){sub j=1}{sup {infinity}} consider a function {gamma}({xi}):=min{sub j=1,2,...}({xi}{psi}{sub j}){sup 1/j}. Conditions for the sequence ({psi}{sub j}){sub j=1}{sup {infinity}} and the operator A are found that ensure the equality lim sup{sub j{yields}}{sub {infinity}}((||A{sup j}f||)/({psi}{sub j})){sup 1/j} = lim sup{sub {xi}}{sub {yields}}{sub {infinity}}{xi}/({gamma}(E{sub {xi}}(f,A){sup -1})) for f element of D{sub {infinity}}(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 {l_brace}{psi}{sub j}{r_brace}{sub j=1}{sup {infinity}}. One consequence of the above formula is an expression in terms of E{sub {xi}}(f,A) for the radius of holomorphy of the vector-valued function F(zA)f, where f element of D{sub {infinity}}(A), and F(z):={sigma}{sub j=1}{sup {infinity}}z{sup j}/{psi}{sub j} is an entire function.

• OSTI ID: 21205651
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 12 Vol. 192; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BANACH SPACE
COMPLEX MANIFOLDS
FUNCTIONS
VECTORS