On sectorial matrices and their inequalities

Assume that the numerical range of T∈Mn is a subset of the sector Sα={z∈C:Re(z)>0,|Im(z)|≤tan⁡(α)Re(z)}. It is proved that, if T=U|T| is the polar decomposition of T, then|T|≤sec⁡(α)[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.