Fuglede–Putnam theorem for \((\alpha , \beta )\)-normal operators

Abstract

In this paper, we obtain some properties of \((\alpha , \beta )\)-normal and we prove following assertions.

  1. (i)

    If T is \((\alpha , \beta )\)-normal operator, S is an invertible operator and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\).

  2. (ii)

    If T is totally \((\alpha , \beta )\)-normal operator, then the range of generalized derivation \(\delta _{T}: \mathcal {B}( {\mathcal H}) \ni X \rightarrow TX- XT \in \mathcal {B}( {\mathcal H}) \) is orthogonal to its kernel.

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Correspondence to A. Bachir.

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Bachir, A., Prasad, T. Fuglede–Putnam theorem for \((\alpha , \beta )\)-normal operators. Rend. Circ. Mat. Palermo, II. Ser 69, 1243–1249 (2020). https://doi.org/10.1007/s12215-019-00454-9

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Keywords

  • Fuglede–Putnam theorem
  • Hyponormal operator
  • \((\alpha</Keyword> <Keyword>\beta )\)-Normal operator
  • Orthogonality

Mathematics Subject Classification

  • 47B20
  • 47A10