Journal of Soviet Mathematics

, Volume 16, Issue 3, pp 1175–1179 | Cite as

Two methods for avoiding hereditary completeness

  • L. N. Dovbysh
  • N. K. Nikol'skii
Article
  • 18 Downloads

Abstract

A family of vectors
of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {xn′;n≥1}((xn,xk′)=δnk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′n)xn:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families
of the same type but already complete in H.. Clearly, in this case
cannot be hereditarily complete. The second method consists in considering linear deformation n :n⩾1 of the orthogonal basesn: n⩾1; here A is an unbounded operator of a special type.

Keywords

Fourier Fourier Series Unbounded Operator Linear Deformation Minimal Family 

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Literature cited

  1. 1.
    N. K. Nikol'skii, “The present state of the spectral analysis-synthesis problem, I.” Trude Letnei Shkoly po Teorii Lineinykh Operatorov (Novosibirsk, 1975), Nauka (1977).Google Scholar
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    A. S. Markus, “The problem of spectral synthesis for operators with point spectrum,” Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 3, 662–688 (1970).MATHMathSciNetGoogle Scholar
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    N. K. Nikol'skii, “Selected problems in weighted approximation and spectral analysis,” Tr. Mat. Inst. im. V. A. Steklova,120, Nauka, Moscow-Leningrad (1974).Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • L. N. Dovbysh
  • N. K. Nikol'skii

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