Abstracts of papers
Bulletin of the American Mathematical Society
The following papers have been submitted to the Secretary and the Associate Secretaries of the Society for presentation at meetings of the Society. They are numbered serially throughout this volume. Cross references to them in the reports of the meetings will give the number of this volume, the number of this issue, and the serial number of the abstract. 203. J. W. Calkin : A quotient ring over the ring of bounded operators in Hilbert space. I. Let 43 denote the ring of bounded everywhere
... d operators in Hilbert space & The subset'G of totally continuous operators is a two-sided ideal in 43 in the ordinary algebraic sense, and the quotient ring 43/5 can be constructed in the usual way; moreover, since T5 is closed with respect to the operation *, this operation can be defined in 43/G too. Defining a norm in 43/G by the equation \a\ = g.l.b.| A |, A in a, where | A | is the bound of the operator A in $, the author shows that 43/G is a complete metric space. Further, he shows that there exists a unitary space 8 (nonseparable, however) and a set VXl of bounded everywhere defined operators in 2 which is a (+, ', *)-isomorphism of 43/Z3. In addition, if T(a) denotes the element of M corresponding to a in 43/Z3, the bound of T(a) is \a\. Thus the correspondence 43/I3<->f7yT is an isometry, and SW is an algebraic ring of operators closed in the uniform topology. Other results are: If 3 is an arbitrary two-sided ideal in43, either 3^Ts> or 3 =43. Every self-adjoint transformation T(a) in Vît has a pure point spectrum. (Received February 24, 1940.) 204. J. W. Calkin: A generalization of a theorem of Weyl.