Polar decompositions and related classes of operators in spaces \(\Pi_\kappa\). (Q1865927)

An invertible selfadjoint operator \(H\) on a separable Hilbert space \(({\mathcal G},\langle.,.\rangle)\) is used to produce an indefinite inner product \([x,y]=\langle Hx, y\rangle\), such that \(({\mathcal G},[.,.])\) becomes a \(\Pi_{\mathcal K}\) space. The \(H\)-polar decomposition of an operator \(X\) is a representation of the form \(X= UA\), where \(A\) is \(H\)-selfadjoint, and \(U\) is an \(H\)-isometry. Using adequate Witt techniques, the authors extend several results concerning the \(H\)-polar decomposition from finite-dimensional spaces to arbitrary Pontryagin spaces. The main theorems, which are based on the Potapov's theory of \(H\)-modulus, state the existence of the selfadjoint square root and logarithm.