Convergence of functions of self-adjoint operators and applications (Q739939)

The main result of this paper is as follows. Let \(f\) be a strictly convex continuous function on an interval \(I\), let \(\mathcal H\) be a Hilbert space, and let \((H_i)\) be a net of bounded self-adjoint operators on \(\mathcal H\) such that \(\sigma(H_i)\subseteq I\) for all \(i\). If \((H_i)\) converges weakly to a bounded self-adjoint operator \(H\) with \(\sigma(H)\subseteq I\), and if \((f(H_i))\) converges weakly to \(f(H)\), then \((\varphi(H_i))\) converges strongly to \(\varphi(H)\) for each bounded continuous function \(\varphi\) on \(I\). In particular, if \((H_i)\) is bounded, then \((H_i)\) converges strongly to \(H\). The assumption that \(f\) is strictly convex is necessary, as is the assumption that \((H_i)\) is bounded in the latter statement. Using the main theorem, a previous result from [the author, Can. J. Math. 40, No. 4, 865--988 (1988; Zbl 0647.46044), Proposition 2.59(a)] is extended from operator convex functions to strictly convex functions, leading to the following statement. Let \(A\) be a \(C^\ast\)-algebra, \(f\) a continuous strictly convex function on an interval \(I\), and let \(h\in A^{\ast\ast}\) be self-adjoint with \(\sigma (h)\subseteq I\). If both \(h\) and \(f(h)\) are quasimultipliers of \(A\), then \(h\) is a multiplier of \(A\). Additional results are obtained, including an extension of [\textit{W. Arveson}, Isr. J. Math. 184, 349--385 (2011; Zbl 1266.46045), Theorem 9.4] to infinite dimensional spaces.