Positive Self-adjoint Operator Extensions with Applications to Differential Operators

In this paper we consider extensions of positive operators. We study the connections between the von Neumann theory of extensions and characterisations of positive extensions via decompositions of the domain of the associated form. We apply the results to elliptic second order differential operators and look in particular at examples of the Laplacian on a disc and the Aharonov–Bohm operator.

Correct selfadjoint and positive extensions of nondensely defined minimal symmetric operators

LetA0be a closed, minimal symmetric operator from a Hilbert spaceℍintoℍwith domain not dense inℍ. LetA^also be a correct selfadjoint extension ofA0. The purpose of this paper is (1) to characterize, with the help ofA^, all the correct selfadjoint extensionsBofA0with domain equal toD(A^), (2) to give the solution of their corresponding problems, (3) to find sufficient conditions forBto be positive (definite) whenA^is positive (definite).

Positive extensions of automorphisms of spin factors

SynopsisA spin factor is a JW-factor of type I2. It is shown that certain automorphisms of finite dimensional spin factors extend to extremal positive linear maps on complex matrix algebras which are not decomposable, and hence, do not preserve extreme rays of the positive cone.