Convergence of functions of self-adjoint operators and applications

TL;DR

This paper establishes that weak convergence of functions of self-adjoint operators combined with weak convergence of the operators themselves implies strong convergence, with applications to invariant subspaces and operator algebra properties.

Contribution

It proves a new convergence criterion linking weak and strong convergence of self-adjoint operators via convex functions, and applies this to invariant subspace and operator algebra problems.

Findings

Weak convergence plus convergence of functions implies strong convergence.

Verification of Arveson's conjecture in infinite dimensions.

Conditions under which operators are multipliers or strongly q-continuous.

Abstract

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.