Norm convergence of sectorial operators on varying Hilbert spaces

Abstract

Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. Many previous contributions to this subject consider either concrete examples of perturbations, or an abstract setting where weak or strong convergence of the resolvents is used. However, it seems that the first results on norm resolvent convergence in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on Hilbert spaces that depend on a parameter. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces. In addition, we show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold's skeleton graph plays a prominent role, being our main application.Mathematics subject classification (2010): 34B45, 35P05, 47D06.

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

“…Both the classical Trotter-Kato-Neveu-like theorems and the more recently developed theory of 'degenerate' convergence to a semigroup that is defined only on a subspace of the original Banach space, have already proved to be instrumental in providing insight into phenomena originating in various fields of pure and applied mathematics. These include, but are not limited to, Markov processes [36], boundary theory for Markov chains [18], mathematical physics [72], and mathematical biology [17]; it is also perhaps worth noting the paper [62] where convergence of semigroups related to quite general boundary conditions is studied. Hence, our investigations may be seen as a reflection of our deep conviction that thin layer approximation, having intrinsic nature of a singular perturbation, could be seen as an integral part of the theory of degenerate convergence of semigroups of operators.…”

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