On a conjecture by Mbekhta about best approximation by polar factors

CHIUMIENTO, EDUARDO

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2021 vol. 149 p. 3913 - 3922

0002-9939

The polar factor of a bounded operator acting on a Hilbert spaceis the unique partial isometry arising in the polar decomposition. It is wellknown that the polar factor might not be a best approximant to its associatedoperator in the set of all partial isometries, when the distance is measured inthe operator norm. We show that the polar factor of an arbitrary operatorT is a best approximant to T in the set of all partial isometries X such thatdim(ker(X)∩ker(T)⊥) ≤ dim(ker(X)⊥∩ker(T)). We also provide a characterizationof best approximations. This work is motivated by a recent conjectureby M. Mbekhta, which can be answered using our results.