Comment on `The tan {\theta} theorem with relaxed conditions', by Y. Nakatsukasa

TL;DR

This paper demonstrates that a key result in Nakatsukasa's recent work on the tan θ theorem is a corollary of a previous theorem, and offers an alternative matrix formulation for another related tan θ theorem.

Contribution

It shows the connection between Nakatsukasa's theorem and earlier results, and provides a new finite-dimensional matrix formulation for related theorems.

Findings

Nakatsukasa's main result is a corollary of Kostrykin et al.'s theorem.

Provides an alternative finite-dimensional matrix formulation.

Clarifies relationships among various tan θ theorems.

Abstract

We show that in case of the spectral norm, one of the main results of the recent paper "The tan {\theta} theorem with relaxed conditions", by Yuji Nakatsukasa, published in Linear Algebra and its Applications is a corollary of the tan {\theta} theorem proven in [V.Kostrykin, K.A.Makarov, and A.K.Motovilov, On the existence of solutions to the operator Riccati equation and the tan {\theta} theorem, IEOT 51 (2005), 121-140]. We also give an alternative finite-dimensional matrix formulation of another tan {\theta} theorem proven in [S.Albeverio and A.K.Motovilov, The a priori tan {\theta} theorem for spectral subspaces, IEOT 73 (2012), 413-430].