Representation and approximation of the polar factor of an operator on a Hilbert space

Let \begin{document}$ H $\end{document} be a complex Hilbert space and let \begin{document}$ \mathcal{B}(H) $\end{document} be the algebra of all bounded linear operators on \begin{document}$ H $\end{document}. The polar decomposition theorem asserts that every operator \begin{document}$ T \in \mathcal{B}(H) $\end{document} can be written as the product \begin{document}$ T = V P $\end{document} of a partial isometry \begin{document}$ V\in \mathcal{B}(H) $\end{document} and a positive operator \begin{document}$ P \in \mathcal{B}(H) $\end{document} such that the kernels of \begin{document}$ V $\end{document} and \begin{document}$ P $\end{document} coincide. Then this decomposition is unique. \begin{document}$ V $\end{document} is called the polar factor of \begin{document}$ T $\end{document}. Moreover, we have automatically \begin{document}$ P = \vert T\vert = (T^*T)^{\frac{1}{2}} $\end{document}. Unlike \begin{document}$ P $\end{document}, we have no representation formula that is required for \begin{document}$ V $\end{document}.

In this paper, we introduce, for \begin{document}$ T\in \mathcal{B}(H) $\end{document}, a family of functions called a "polar function" for \begin{document}$ T $\end{document}, such that the polar factor of \begin{document}$ T $\end{document} is obtained as a limit of a net built via continuous functional calculus from this family of functions. We derive several explicit formulas representing different polar factors. These formulas allow new for methods of approximations of the polar factor of \begin{document}$ T $\end{document}.