We show that to each symmetric elliptic operator of the formon a bounded Lipschitz domain Ω ⊂ R d one can associate a self-adjoint Dirichlet-to-Neumann operator on L 2 (∂Ω), which may be multi-valued if 0 is in the Dirichlet spectrum of A. To overcome the lack of coerciveness in this case, we employ a new version of the Lax-Milgram lemma based on an indirect ellipticity property that we call hidden compactness. We then establish uniform resolvent convergence of a sequence of Dirichlet-to-Neumann operators whenever their coefficients converge uniformly and the second-order limit operator in L 2 (Ω) has the unique continuation property. We also consider semigroup convergence.April 2013.

We study the regular part of a densely defined sectorial form, first in the abstract setting and then under mild conditions for a differential sectorial form. The regular part of the latter turns out to be again a differential sectorial form. Moreover, we characterize when taking the real part of a differential sectorial form commutes with taking the regular part. An example shows that these two operations do not commute in general.

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator A in a separable Hilbert space H is unitarily equivalent to an operator B in H such that D(A) ∩ D(B) = {0}. Equivalently this can be formulated as a property for nonclosed operator ranges. We will show that von Neumann's theorem does not directly extend to the nonseparable case.In this paper we prove a characterisation of the property that an operator range R in a general Hilbert space H admits a unitary operator U such that U R ∩ R = {0}. This allows us to study stability properties of operator ranges with the aforementioned property.

Abstract. The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. M c Intosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms.Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

Abstract. We present a formula for the regular part of a sectorial form that represents a general linear second-order differential expression that may include lower-order terms. The formula is given in terms of the original coefficients. It shows that the regular part is again a differential sectorial form and allows to characterise when also the singular part is sectorial. While this generalises earlier results on pure second-order differential expressions, it also shows that lower-order terms truly introduce new behaviour.