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On (A, m)-Symmetric Operators in a Hilbert Space

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Abstract

For a positive \(A\in \mathcal {B}(\mathbb {H})\), an operator \(T\in \mathcal {B}(\mathbb {H})\) is said to be (A, m)-symmetric if it satisfies the operator equation \(\displaystyle \sum \nolimits _{k=0}^{m}(-1)^{m-k} \;{m\atopwithdelims ()k}T^{*k}AT^{m-k}=0.\) This class of operators seems a natural generalization of m-symmetric operators on a Hilbert space. In this paper, first we give various properties related to such a family. Then, we prove that if T and Q are commuting operators, T is (A, m)-symmetric and Q is l-nilpotent, then \((T+Q)\) is \((A,m + 2l - 2)\)-symmetric. In addition, we show that every power of an (A, m)-symmetric operator is also (A, m)-symmetric. Some connection between (A, m)-symmetric operators and \(C_0\)-semigroups are also shown. Finally, we characterize the spectra of such operators.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, 6072, Gabés, Tunisia

    N. Jeridi & R. Rabaoui

  2. Laboratory of Mathematical Analysis and Applications (LMAA-LR11-ES11), Faculty of Mathematical, Physical and Natural Sciences of Tunis, University of Tunis El-Manar, Tunis, Tunisia

    R. Rabaoui

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  1. N. Jeridi
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  2. R. Rabaoui
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Jeridi, N., Rabaoui, R. On (A, m)-Symmetric Operators in a Hilbert Space. Results Math 74, 124 (2019). https://doi.org/10.1007/s00025-019-1049-0

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