Effect Algebras of Positive Self-adjoint Operators Densely Defined on Hilbert Spaces

Z Riečanová

doi:10.14311/1412

Abstract

We show that (generalized) effect algebras may be suitable very simple and natural algebraic structures for sets of (unbounded) positive self-adjoint linear operators densely defined on an infinite-dimensional complex Hilbert space. In these cases the effect algebraic operation, as a total or partially defined binary operation, coincides with the usual addition of operators in Hilbert spaces.

Highlights

For any linear operator A densely defined on a Hilbert space H one can define its adjoint operator A∗
Differential operators form a class of unbounded operators
The Laplace operator is an example of an unbounded positive linear operator

Summary

Introduction

For any linear operator A densely defined on a Hilbert space H one can define its adjoint operator A∗. The set of all positive self-adjoint linear operators densely defined in H will be denoted by Sp(H).

Objectives

Results

Conclusion