Properties of two unitary operator functions involving idempotents

Abstract

Let P be an idempotent operator on a Hilbert space \(\mathcal {H}.\) We denote two unitary operator functions \(U_{\lambda }\) and \(V_{\lambda }\) by

$$\begin{aligned} U_{\lambda }:=(\lambda P+I)|\lambda P+I|^{-1} \hbox { } \hbox { and }\hbox { } V_{\lambda }:=(\lambda P^{*}+I)|\lambda P^{*}+I|^{-1}, \ \ \hbox { for }\lambda \in \mathbb {C}\backslash \{-1\}. \end{aligned}$$

In this paper, we first give the specific structures of \(U_{\lambda }\) and \(V_{\lambda },\) respectively. Then the sufficient and necessary conditions under which \(U_{\lambda }\) and \(V_{\lambda }\) are symmetries are presented. Moreover, the specific structures and spectra of the unitary operator \(U=\lim \limits _{\lambda \rightarrow -1^+}U_{\lambda }\) are characterized.

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Acknowledgements

The authors would like to express their heart-felt thanks to the anonymous referees for some valuable comments. This work was supported by NSF of China (Nos: 11671242, 11571211) and the Fundamental Research Funds for the Central Universities (GK201801011)

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Correspondence to Yuan Li.

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Communicated by Ilya Spitkovsky.

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Zhang, J., Niu, J. & Li, Y. Properties of two unitary operator functions involving idempotents. Ann. Funct. Anal. 11, 540–554 (2020). https://doi.org/10.1007/s43034-019-00036-x

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Keywords

  • Idempotents
  • Symmetries
  • The operator function

Mathematics Subject Classification

  • 47A05
  • 47A62
  • 46C20