Approximating classes of sequences: The Hermitian case

Given a sequence {An} of matrices An of increasing dimension dn with dk>dq for k>q, k,q∈N, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, whenever natural and mild conditions are satisfied. In this note we focus our attention on the Hermitian case and we show that is an a.c.s. for {f(An)}, if is an a.c.s. for {An}, {An} is sparsely unbounded, and f is a suitable continuous function defined on R. We also discuss the potential impact and future developments of such a result.