Numerical radius inequalities of sectorial matrices

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 xn complex matrix with the numerical range W( A) satisfying W( A) subset of {re(+/- i theta) : theta(1) <= theta <= theta(2)}, where r > 0 and theta(1), theta(2). [0, pi/2], then (i) w(A) >= csc gamma/2 ||A|| + csc gamma/2 | ||(sic) (A)|| - ||(sic) (A)|| |, and (ii) w(2) (A) >= csc(2)gamma/4 || AA * + A* A || + csc(2)gamma/2 | ||(sic) (A)||(2) -||(sic) (A)||(2)|, where gamma = max{theta(2), pi/2 - theta(1)}. We also prove that if A, B are sectorial matrices with sectorial index gamma is an element of [0, pi/2) and they are double commuting, then w( AB) = <=(1 + sin(2)gamma) w( A) w(B).