Vol. 18, No. 3, 1966

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Everywhere defined linear transformations affiliated with rings of operators

Ernest Lyle Griffin, Jr.

Vol. 18 (1966), No. 3, 489–493

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 = Mthen T is bounded.

Mathematical Subject Classification
Primary: 46.65
Received: 16 April 1965
Published: 1 September 1966
Ernest Lyle Griffin, Jr.