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Lorentz sequence spaces

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Abstract

It is shown that the condition

$$\mathop {\sup }\limits_n \{ n^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \left( {\sum\nolimits_{j \leqslant n} {c_j^2 } } \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} /\sum\nolimits_{j \leqslant n} {c_j } \}< \infty $$

on the normalizing sequence {cj}j<∞ of the Lorentz sequence space Λ(c) is a necessary and sufficient condition for having each bounded linear operator acting from an arbitrary ℒ ∝ -space into Λ(c) be 2-absolutely summing.

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Literature cited

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Authors and Affiliations

  1. Khar'kov Institute of Communal Construction Engineers, USSR

    S. A. Rakov

Authors
  1. S. A. Rakov
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Additional information

Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 501–510, October, 1976.

The author thanks M. I. Kadets for posing the problem and for his guidance.

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Rakov, S.A. Lorentz sequence spaces. Mathematical Notes of the Academy of Sciences of the USSR 20, 837–842 (1976). https://doi.org/10.1007/BF01098899

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  • DOI: https://doi.org/10.1007/BF01098899

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