Generalized powers and generalized logarithms of operators

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 th...

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Bibliographic Details
Published in:Rendiconti del Circolo matematico di Palermo 2023-12, Vol.72 (8), p.3829-3840
Main Author: Hranislav, Stanković
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Geometry
Linear operators
Logarithms
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
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Summary: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 ↦ A T . We also introduce a notion of generalized logarithms. More precisely, for two positive and invertible operators A and B such that 1 ∉ σ ( A ) , we define the logarithm of B to base A , denoted by log A B , and investigate some of its properties.
ISSN:0009-725X
1973-4409
DOI:10.1007/s12215-023-00867-7