On Subnormality and Formal Subnormality for Tuples of Unbounded Operators

We introduce the notion of (joint) formal normality for a collection of unbounded linear operators on a separable Hilbert space H which is, in some sense, a natural generalization of the notion of formal normality for a single operator. We give some relations between this new notion and (joint) subnormality and hyponormality. We adapt, in particular, a proof of Stochel and Szafraniec to give necessary and sufficient conditions for a tuple of unbounded operators with invariant domain to be (jointly) subnormal.