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Numerical radius inequalities of sectorial matrices

Author : Dr Pintu Bhunia

Year : 2023

Publisher : Birkhauser

Source Title : Annals of Functional Analysis

Document Type :

Abstract

We obtain several upper and lower bounds for the numerical radius of sectorial matrices. We also develop several numerical radius inequalities of the sum, product and commutator of sectorial matrices. The inequalities obtained here are sharper than the existing related inequalities for general matrices. Among many other results we prove that if A is an n× n complex matrix with the numerical range W(A) satisfying W(A)⊆{re±iθ:θ1≤θ≤θ2}, where r> 0 and θ1, θ2∈ [ 0 , π/ 2 ] , then (i)w(A)≥cscγ2‖A‖+cscγ2|‖ℑ(A)‖-‖ℜ(A)‖|,and(ii)w2(A)≥csc2γ4‖AA∗+A∗A‖+csc2γ2|‖ℑ(A)‖2-‖ℜ(A)‖2|, where γ= max { θ2, π/ 2 – θ1} . We also prove that if A, B are sectorial matrices with sectorial index γ∈ [ 0 , π/ 2 ) and they are double commuting, then w(AB) ≤ (1 + sin 2γ) w(A) w(B).