  The main result (roughly) is that if (H_i) converges weakly to H and if also
f(H_i) converges weakly to f(H), for a single strictly convex continuous
function f, then (H_i) must converge strongly to H. One application is that if
f(pr(H)) = pr(f(H)), where pr denotes compression to a closed subspace M, then
M must be invariant for H. A consequence of this is the verification of a
conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite
dimensional case. And there are two applications to operator algebras. If h and
f(h) are both quasimultipliers, then h must be a multiplier. Also (still
roughly stated) if h and f(h) are both in pA_sa p, for a closed projection p,
then h must be strongly q-continuous on p.
