G. Epifanio

V*‐algebras: A particular class of unbounded operator algebras

We consider a weak unbounded commutant for a set of unbounded operators and we examine op*‐algebras which coincide with some of their bicommutant. This class of op*‐algebras, called V*‐algebras, shows some properties close to those which hold true for bounded operator algebras.

Partial *‐algebras and Volterra convolution of distribution kernels

The set of distribution kernels is considered as a partial *‐algebra when the multiplication of two kernels is defined as their Volterra convolution. The partial *‐algebra of distribution kernels corresponding to a certain partial *‐algebra of closable operators in L2(Ω) is studied.

Partial *‐algebras of matrices and operators

The set M∞ of infinite matrices and the set Q∞ of squarable matrices are considered as partial *‐algebras. The connection between Q∞ and two partial *‐algebras of closed operators is studied. Conditions for a matrix representation in ‘‘von Neumann’s sense’’ of a family of closed operators are given.

On the matrix representation of unbounded operators

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.

Some topics in pre‐Hilbert space

We examine some questions concerning pre‐Hilbert spaces and operators defined in them. Some classical results, which hold true in Hilbert space, are extended, under particular conditions, to noncomplete space.

Some spectral properties in algebras of unbounded operators

Some aspects of spectral theory in algebras of unbounded operators are studied. After having pointed out the pathologies of the spectral behavior of these operators we give a sufficient condition in order that a self‐adjoint operator admit a spectral decomposition with spectral measure with values in the same algebra. Some examples illustrating the developed ideas are given.

On the matrix representation of closed linear operators: An extension of the Von Neumann’s theory

The general theory of the matrix representation of operators in scalar product space is examined. It is proved that an extension of the Von Neumann’s theory on the matrix representation of closed symmetric operators in Hilbert space is possible for a larger class of closed operators. A necessary and sufficient condition for the existence of a matrix representation of operators in ’’Von Neumann’s sense’’ is given.

Complete sets of non‐self‐adjoint observables: An unbounded approach

The notion of completeness of a set S of compatible observables represented by maximal symmetric operators is discussed directly in terms of unbounded operators. In contrast with what happens for self‐adjoint observables, the present framework forces us to involve some partial algebraic structures such as the partial GW*‐algebra generated by S. In this way the previous approaches based on von Neumann algebras and on O*‐algebras are generalized.

Remarks on a theorem by G. Epifanio, ’’On the matrix representation of unbounded operators’’ [J. Math. Phys. 17, 1688 (1976)]

We give a generalization of a theorem concerning the change of basis for the matrix representation of unbounded operators defined in a scalar product space. We introduce for the proof a suitable structure which can be useful when one has to make operations with operators defined between different scalar product spaces.