  In this work we introduce a new measure for the dispersion of the spectral
scale of a Hermitian (self-adjoint) operator acting on a separable infinite
dimensional Hilbert space that we call spectral spread. Then, we obtain some
submajorization inequalities involving the spectral spread of self-adjoint
operators, that are related to Tao's inequalities for anti-diagonal blocks of
positive operators, Kittaneh's commutator inequalities for positive operators
and also related to the Arithmetic-Geometric mean inequality. In turn, these
submajorization relations imply inequalities for unitarily invariant norms (in
the compact case).
