On a conjecture by Mbekhta about best approximation by polar factors

The polar factor of a bounded operator acting on a Hilbert space is the unique partial isometry arising in the polar decomposition. It is well known that the polar factor might not be a best approximant to its associated operator in the set of all partial isometries, when the distance is measured in the operator norm. We show that the polar factor of an arbitrary operator $T$ is a best approximant to $T$ in the set of all partial isometries $X$ such that $\dim (\ker(X)\cap \ker(T)^\perp)\leq \dim (\ker(X)^\perp\cap \ker(T))$. We also provide a characterization of best approximations. This work is motivated by a recent conjecture by M. Mbekhta, which can be answered using our results.