A note on sectorial matrices
Mohammad Alakhrass · Linear and Multilinear Algebra · 2019
Mohammad Alakhrass* Department of Mathematics, University of Sharjah, Sharjah, UAECONTACT Mohammad Alakhrass [email protected] of Mathematics, University of Sharjah, Sharjah 27272, UAEABSTRACTAssume 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)}, for some α∈[0,π/2). It is proved that |T|≤sec(α)2Re(T)+U∗Re(T)U, for some unitary U∈Mn. As a consequence, we prove the following singular value inequalities sj(T)≤sec(α)s[(j+1)/2](Re(T))for j=1,2,…,n, where [x] is the greatest integer ≤x. In the case where T∈M2n partitioned as T=T11T12T21T22,whereTij∈Mn, i,j=1,2, the following log-majorization inequality is proved ∏l=1ksl(Tij)≤seck(α)∏l=1ksl1/2(Re(Tii))sl1/2(Re(Tjj)),i,j=1,2, for k=1,2,…,n. As a result, we get the following Hölder type inequality ∥|T12|r∥≤secr(α)∥T11rp/2∥1/p∥T22rq/2∥1/q, for any unitarily invariant norm ∥⋅∥. Here, r,p and q are positive numbers such that 1/p+1/q=1. Related inequalities are also proved.