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The closed operators in a Hilbert space H are characterized as quotients AB ~l of continuous operators on H such that the vector sum A*(H)+ B*(H) is closed. This leads to the function T(A) = A(\- A*A)~1/2, which is shown to map the strictly contractive operators on H reversibly onto the closed densely-defined operators, so as to preserve the selfadjoint and nonnegative conditions. Introduction. Suppose that (77, ") denote the usual product space with corresponding norm || • ||". By a closed operator in 77 is meant a linear function from a linear subspace of 77 into H which is closed in 77 X 77. Let % (H) denote the algebra of all continuous linear functions from 77 into 77 ("including" the complex numbers) and for each A in % (77), let A * denote the adjoint of A, let A ~ ' denote the inverse of the restriction of A to the closure of A*(H), and-in case A is nonnegative-let A~x/2 denote L41//2)_1. Note that for each A in % (77), A ~ XA is the orthogonal projection from H onto the closure of A *(H).

Representing a closed operator as a quotient of continuous operators