A note on sectorial matrices

Assume that the numerical range of T. Mn 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)}, for some a is an element of [0, pi/2). It is proved that vertical bar T vertical bar <= sec(alpha)/2[Re(T) + U* (Re(T)) U], for some unitary U is an element of M-n. As a consequence, we prove the following singular value inequalities s(j)(T) <= sec(alpha) s[(j+1)/2](Re(T)) for j = 1, 2, ... , n, where [x] is the greatest integer <= x. In the case where T is an element of M-2n partitioned as T = ((T11)(T21) (T12)(T22)), where T-ij is an element of M-n, i, j = 1, 2, the following log-majorization inequality is proved Pi(k)(l=1) s(l)(T-ij) <= sec(k)(alpha) Pi(k)(l=1) s(l)(1/2) (Re(T-ii))s(l)(1/2) (Re(T-jj)), i, j = 1, 2, for k = 1, 2, ... , n. As a result, we get the following Holder type inequality parallel to vertical bar T-12 vertical bar(r)parallel to = sec(r)(alpha)parallel to T-11(rp/2)parallel to(1/p)parallel to T-22(rq/2)parallel to(1/q), for any unitarily invariant norm parallel to center dot parallel to. Here, r,p and q are positive numbers such that 1/p + 1/q = 1. Related inequalities are also proved.