Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space

Abstract

The developments of the present paper center around the observation that the ring B of bounded everywhere defined operators in Hilbert space contains non-trivial two-sided ideals.' This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and at the same time differentiates 93 sharply from the ring of all linear operators over a unitary space with finite dimension number. As examples of two-sided ideals in 93 we may mention here the class of all operators A such that J(A), the range of A, has a finite dimension number, the class of all operators of Hilbert-Schmidt type,2 and the class fT of all totally continuous operators. Except for the ideal (0), every two-sided ideal in 93 contains the first ideal mentioned, and except for the ideal J3 itself, every twosided ideal in 93 is contained in the ideal f. Moreover, on the basis of the special spectral properties of the self-adjoint members of sJ, it is possible to characterize every two-sided ideal in ?B very simply in terms of the spectra of its nonnegative self-adjoint elements; for both the formulation and the proof of this result, which together with the facts mentioned above is discussed in ?1, the author is indebted to J. v. Neumann. The restriction of our attention to those ideals in if which are two-sided is basic for the points which we wish to develop; the two-sidedness compensates for the absence of commutativity in 93 in such a way as to permit the construction of quotient rings by the standard methods of abstract algebra.3 These rings, which are of course homomorphs of B with respect to addition and multiplication, are also homomorphs of. @ with respect to the operation *, and exhibit all of the formal properties of matrix algebras. This is established in ?2, and there also various properties of the associated congruences in gB are discussed. The remainder of the paper deals solely with the quotient ring @/{, where fJ is the ideal of totally continuous operators, and the associated congruence in i3. For essentially topological reasons, this is the only one of the quotient rings in