Abstract
Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space \({\mathcal{H}}\) into a Hilbert space \({\mathcal{K}}\) are defined, motivated by the inequality \({|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}}\) . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on \({\mathcal{H}}\) and a bounded Borel function \({f: \mathbb{R}_+ \to \mathbb{R}_+}\) , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Niemiec, P. Generalized Absolute Values and Polar Decompositions of a Bounded Operator. Integr. Equ. Oper. Theory 71, 151–160 (2011). https://doi.org/10.1007/s00020-011-1896-x
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DOI: https://doi.org/10.1007/s00020-011-1896-x