Convergence of functions of self-adjoint operators and applications
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.
Keywords
Cite
@article{arxiv.1410.6800,
title = {Convergence of functions of self-adjoint operators and applications},
author = {Lawrence G. Brown},
journal= {arXiv preprint arXiv:1410.6800},
year = {2017}
}
Comments
This paper is intended for a dual audience. Section 2, which contains the main result, is suitable for a general audience. Section 3 concerns technical operat or algebraic questions, though I have tried to present it with a minimum of tech nicality. As with all my papers, I welcome comments, especially for this paper, since I know almost nothing about Korovkin type theorems