Operators in divergence form and their Friedrichs and Krein extensions

Abstract

Abstract. For a densely defined nonnegative symmetric operator A = L * 2 L1 in a Hilbert space, constructed from a pair L1 ⊂ L2 of closed operators, we give expressions for the Friedrichs and Kreȋn nonnegative selfadjoint extensions. Some conditions for the equality (L * 2 L1) * = L * 1 L2 are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given.

Around the Van Daele–Schmüdgen Theorem

“…0 for all f ∈ dom A, the operator A is non-negative. This kind of operators A we call the operators in divergence form [7]. The next assertions are established in [7].…”

“…This kind of operators A we call the operators in divergence form [7]. The next assertions are established in [7]. Theorem 2.7.…”

Some Remarks on the Operator T^*T

“…Kovalev [1, Theorem 1.1, Proposition 3.3], [2, Theorems 1.1, 1.2]. We decided to keep this section in this archive submission as the approach in Theorem 3.1 below differs a bit from that in [1] and [2]. However, we emphasize that papers [1] and [2] go far beyond the scope of this section, in particular, they also discuss the Krein-von Neumann extension in addition to the Friedrichs extension.…”

The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

“…elliptic second order differential operators on Euclidean domains, the Friedrichs extension is a very natural object; for instance, for the minimal symmetric Laplacian on a bounded domain in R n corresponding to both Dirichlet and Neumann boundary conditions, the Friedrichs extension is the self-adjoint Laplacian subject to Dirichlet boundary conditions. On the other hand, in the same setting, the Krein-von Neumann extension corresponds to certain non-local boundary conditions which can be described in terms of the associated Dirichlet-to-Neumann map; for properties of the Krein-von Neumann extension of elliptic differential operators and recent related developments, we refer the reader to [4,9,11,23,24,37].…”