On sectorial matrices and their inequalities

Assume that the numerical range of T is an element of M-n is a subset of the sector S-alpha = {z is an element of C : Re(z) > 0, vertical bar Im(z)vertical bar <= tan(alpha)Re(z)}. It is proved that, if T = U vertical bar T vertical bar is the polar decomposition of T, then vertical bar T vertical bar <= sec(alpha) [Re(T)#U*Re(T)U]. Several consequences of this inequality are established, including singular value inequalities, unitarily invariant norm inequalities and sharp determinant inequalities. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This article discusses the restrictions on the numerical range of T under the condition of its polar decomposition, and provides related inequality conclusions.