Generalized Absolute Values and Polar Decompositions of a Bounded Operator
Piotr Niemiec · Integral Equations and Operator Theory · 2011
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.