Generalized absolute values and polar decompositions of a bounded operator
Let \(\mathcal{H}\) and \(\mathcal{K}\) be Hilbert spaces and \(T:\mathcal{H}\to\mathcal{K}\) a bounded linear operator. Then \(T=Q|T|\), where \(|T|=\sqrt{T^*T}\) and \(Q\) is a partial isometry with \(\mathcal{N}(Q)=\mathcal{N}(T)\), is a classical polar decomposition of \(T\). There are also two other representations given by \(T=|T^*|Q=\sqrt{|T^*|}Q\sqrt{|T|}\). Let \(\mathcal{S}(T)\) consists of all pairs \((A,B)\) such that \(A:\mathcal{H}\to\mathcal{H}\) and \(B:\mathcal{K}\to\mathcal{K}\) are bounded and nonnegative operators and satisfy the inequality \[ |\langle Tx,y\rangle_{\mathcal{K}}|^2\leq\langle Ax,x\rangle_{\mathcal{H}}\langle By,y\rangle_{\mathcal{K}} \] for all \(x\in\mathcal{H}\) and all \(y\in\mathcal{K}\). The pair \((|T|,|T^*|)\) is minimal in the set \(\mathcal{S}(T)\) with respect to the coordinatewise partial order. An operator \(A\) is said to be a generalized absolute value of \(T\) if there is a minimal pair in \(\mathcal{S}(T)\) whose first entry is \(A\). It is shown that, for every generalized absolute value \(A\) of \(T\), there is a unique partial isometry \(Q_A\) and an operator \(A^\#\) such that \[ T=\sqrt{A^\#}Q_A\sqrt{A}. \] The author also shows 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.
scientific article; zbMATH DE number 5982596
scientific article