Let M be a ring of operators on
a Hilbert space H. This paper considers conditions under which an operator T
affiliated with M is bounded (or can be unbounded). In particular, we consider
operators whose domain is the entire space H. We prove: Theorem 3. If M has no
type I factor part, then T is bounded. Theorem 4. T is bounded if and only if T is
bounded on each minimal projection in M. Theorem 6. In order that every
linear mapping from H into H which commutes with M be bounded, it
is necessary and sufficient that M should contain no minimal projection
whose range is an infinite dimensional subspace of H. These results were
suggested by a theorem of J. R. Ringrose: Theorem 8. If M = M′ then T is
bounded.
