Abstract
In a recent paper, Bachir et al. (Rendiconti del Circolo Matematico di Palermo, 2022. https://doi.org/10.1007/s12215-022-00823-x) introduced generalized powers of linear operators. In other words, operators are not raised to numbers but to other operators. They gave several properties as regards this notion. In this paper, we further extend their results and also answer the question regarding the monotonicity of the map \(T\mapsto A^T\). We also introduce a notion of generalized logarithms. More precisely, for two positive and invertible operators A and B such that \(1\notin \sigma (A)\), we define the logarithm of B to base A, denoted by \(\log _AB\), and investigate some of its properties.
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References
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Stanković, H. Generalized powers and generalized logarithms of operators. Rend. Circ. Mat. Palermo, II. Ser 72, 3829–3840 (2023). https://doi.org/10.1007/s12215-023-00867-7
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DOI: https://doi.org/10.1007/s12215-023-00867-7
Keywords
- Generalized powers of operators
- Exponentials of operators
- Generalized logarithms of operators
- Self-adjoint operators
- Positive operators
- Invertible operators