Representing a closed operator as a quotient of continuous operators
William E. Kaufman · Proceedings of the American Mathematical Society · 1978
The closed operators in a Hilbert space H are characterized as quotients A B − 1 A{B^{ - 1}} of continuous operators on H such that the vector sum A ∗ ( H ) + B ∗ ( H ) {A^\ast }(H) + {B^\ast }(H) is closed. This leads to the function Γ ( A ) = A ( 1 − A ∗ A ) − 1 / 2 \Gamma (A) = A{(1 - {A^\ast }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.