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Is there any nontrivial compact generalized shift operator on Hilbert spaces?

  • Fatemah Ayatollah Zadeh Shirazi
  • Fatemeh Ebrahimifar
Article
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Abstract

In the following text for cardinal number \(\tau >0\), and self-map \(\varphi :\tau \rightarrow \tau \) we show the generalized shift operator \(\sigma _\varphi (\ell ^2(\tau ))\subseteq \ell ^2(\tau )\) (where \(\sigma _\varphi ((x_\alpha )_{\alpha<\tau })=(x_{\varphi (\alpha )})_{\alpha <\tau }\) for \((x_\alpha )_{\alpha <\tau }\in {{\mathbb {C}}}^\tau \)) if and only if \(\varphi :\tau \rightarrow \tau \) is bounded and in this case Open image in new window is continuous, consequently Open image in new window is a compact operator if and only if \(\tau \) is finite.

Keywords

Compact operator Generalized shift Hilbert space 

Mathematics Subject Classification

46C99 

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • Fatemah Ayatollah Zadeh Shirazi
    • 1
  • Fatemeh Ebrahimifar
    • 1
  1. 1.Faculty of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

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