Abstract
A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if .In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.
Highlights
Self-adjoint operators on a Hilbert space is an old research topic and very important in many applications
It can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph
In this research note we give a generalization of the Stone Lemma and another equivalent properties for self-adjointness of operators
Summary
Self-adjoint operators on a Hilbert space is an old research topic and very important in many applications. In this research note we give a generalization of the Stone Lemma and another equivalent properties for self-adjointness of operators. There are recent generalizations of classical theorems like the von Neumann theorem which states that for a linear operator , the operator ∗ is self-adjoint (see in [4]) and more and more basic results of functional analysis are formulated on linear relation language. The generalization of operator theory results – like the von Neumann Theorem and the Stone lemma – on linear relation language has a fairly young history. Our goal in this research note is to generalise some basic statements on self-adjoint operators for linear relations. We give a more general Stone lemma and formulate an equivalent condition for self-adjointness of a linear operator on a Hilbert space.
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