Norm convergence of sectorial operators on varying Hilbert spaces (Q2873578)

Let \((A_\varepsilon)_{\varepsilon \geq 0}\) be a family of \(m\)-sectorial operators acting on a family of Hilbert spaces \((H_\varepsilon)_{\varepsilon \geq 0}\), with associated forms \((a_\varepsilon)_{\varepsilon \geq 0}\) and form domains \((V_\varepsilon)_{\varepsilon \geq 0}\). The authors define convergence of \((a_\varepsilon)_{\varepsilon \geq 0}\) to \(a_0\) in norm as a generalisation of convergence on a single Hilbert space in the sense that \(\|a_\varepsilon -a_0\| \to 0\) in the operator norm on the space of sesquilinear forms on \(V_0\). It is shown that this implies norm convergence of operators \(\varphi(A_\varepsilon)\) in a subclass of the holomorphic functional calculus of \(A_\varepsilon\) to \(\varphi(A_0)\), spectral convergence and invariance of subspaces. In particular, this implies norm convergence of the semigroups \((e^{-tA_{\varepsilon}})_{\varepsilon \geq 0}\) to \(e^{-tA_0}\), for fixed \(t\) in the common sector analyticity, and extrapolation results for limits of \(L^p\) contractive families of semigroups. The main motivation of the article is the consideration of manifolds shrinking to a metric graph, as considered previously by \textit{D. Grieser} [Proc. Lond. Math. Soc. (3) 97, No. 3, 718--752 (2008; Zbl 1183.58027)] and \textit{C. Cacciapuoti} and \textit{D. Finco} [Asymptotic Anal. 70, No. 3--4, 199--230 (2010; Zbl 1227.35238)]. For suitably constructed manifolds with boundary, the authors show convergence of the corresponding Laplacians to a Laplacian on the metric graph, and spectral convergence thereof.