Unitary approximation
Catherine L. Olsen · Journal of Functional Analysis · 1989
A unitary approximant for a bounded linear operator T on a separable Hilbert space H is a unitary operator on H of minimum distance from T. In this paper, unitary approximants for certain Fredholm and semi-Fredholm operators of nonzero index are explictly constructed. This completes the answer to the question: which Hilbert space operators have a unitary approximant? The answer is: any operator of index zero, and any operator of nonzero index whose distance from the unitary group is greater than one, and no others. This and related questions are also completely answered in an arbitrary von Neumann algebra. For each element of the algebra, the distance to the invertible group and to the unitary group of the algebra is computed: these distances are related. It is shown that an element has no unitary approximant in the algebra if and only if it has nonzero (relative) index and it is a limit of invertibles in the algebra.