On the matrix representation of unbounded operators

Abstract

It is shown that a matrix representation with properties analogous to the ones that hold for the bounded operators in Hilbert space is possible also for important sets of unbounded operators. These sets consist of the --algebras CD of the linear operators on any noncomplete scalar product space D, which have an adjoint in D. (These algebras have already been studied by the author, in collaboration with others, in previous papers.) Specifically it is proved that for these operators a matrix representation is possible with respect to an arbitrary orthonormal basis within D, in contrast to the situation that has been found by von Neumann for the unbounded closed symmetric operators. The matrix representation of the operators considered here also allows the usual algebraic operations. Besides, the changes of basis induced by automorphisms of D are allowed.