The spectral spread of Hermitian matrices

Norm inequalities for the spectral spread of Hermitian operators

“…Eqs. ( 8), ( 9) and ( 11)) were previously proved for the spectral spread of matrices in [24]. As we explain in the Appendix, the spectral spread of matrices is different from the spectral spread of compact operators, so we need new proofs.…”

“…It turns out that all these singular value inequalities fail in this more general setting. Motivated by our previous work [24,25] (in the finite dimensional case) we first introduce a new notion that we call the spectral spread of a self-adjoint operator that lies in the algebra A = K(H) + C I formed by compact perturbations of multiples of the identity operator acting on H. Then, we obtain inequalities in terms of submajorization (which can be regarded as inequalities for unitarily invariant norms) with respect to the spectral spread, in the general context of compact self-adjoint operators. We regard these new inequalities as natural substitutes of the singular value inequalities mentioned above (that are stronger, but only valid for positive compact operators).…”

“…Indeed, in Section 3 we give new proofs of the infinite dimensioinal version of two inequalities proved in [24]: we show that given a self-adjoint compact operator F ∈ K(H⊕K) represented as a block matrix…”

“…This inequality is not known even for matrices (we give a proof for this case in the Appendix). Using (10) we give a proof of the infinite dimensional version of an inequality proved in [24]: given self-adjoint compact operators C, D ∈ K(H) and a compact operator X ∈ K(H), we have that…”

“…In the Appendix we also discuss, in some detail, versions of these inequalities related with the spectral spread of self-adjoint matrices introduced in [24].…”