Weak commutation relations of unbounded operators and applications

Abstract

Four possible definitions of the commutation relation [S, T] = 11 of two closable unbounded operators S, T are compared. The weak sense of this commutator is given in terms of the inner product of the Hilbert space H where the operators act. Some consequences on the existence of eigenvectors of two numberlike operators are derived and the partial O*-algebra generated by S, T is studied. Some applications are also considered.