Some inequalities involving Hilbert-Schmidt numerical radius on 2 x 2 operator matrices

We present some inequalities related to the Hilbert-Schmidt numerical radius of 2 x 2 operator matrices. More precisely, we present a formula for the Hilbert-Schmidt numerical radius of an operator as follows: w2(T) = sup ?2+?2=1 ||?A + ?B||2, where T = A + iB is the Cartesian decomposition of T ? HS(H).

Singular value inequalities for real and imaginary parts of matrices

Let A = Re A + i Im A be the Cartesian decomposition of square matrix A of order n with Re A = A+A*/2 and Im A = A-A*/2i. Fan-Hoffman?s result asserts that ?j(ReA)? sj(A), j=1,..., n, where ?j(M) and sj(M) stand for the jth largest eigenvalue of M and the jth largest singular value of M, respectively. We investigate singular value inequalities for real and imaginary parts of matrices and prove the following inequalities: sj(Re A) ? 1/4 sj([(|A|+|A*|)-(A+A*)]?[(|A|+|A*|)+(A+A*)]), and sj(Im A) ? 1/4 sj([(|A|+|A*|)- i(A*-A)] ? [(|A|+|A*|) + i(A*-A)]), j = 1,..., n. In particular, we have sj(Re A) ? 1/2 sj((|A|+|A*|)? (|A|+|A*|)), and sj(Im A) ? 1/2 sj((|A|+|A*|)? (|A|+|A*|)), j=1,..., n. Moreover, we also show that these inequalities are sharp.

-TRANSITIVE DIGRAPHS PRESERVING A CARTESIAN DECOMPOSITION

In this paper, we combine group-theoretic and combinatorial techniques to study $\wedge$-transitive digraphs admitting a cartesian decomposition of their vertex set. In particular, our approach uncovers a new family of digraphs that may be of considerable interest.