Let H be a self-adjoint
operator with spectral measure E(S) over the Borel sets S of the real line.
The spectrum of H is said to be strongly concentrated on S if whenever
Hn converges strongly to H in the generalized sense it is true that En(S)
converges strongly to the identity. Sufficient conditions on H are given for
this to occur for a given arbitrary Borel set S and necessary and sufficient
conditions when S is the spectrum of H. In addition several more workable
sufficient conditions are cited and a few examples illustrating the results are
given.
