Approximate Unitary Equivalence to Skew Symmetric Operators

Abstract

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called skew symmetric if \(T\) can be represented as a skew symmetric matrix relative to some orthonormal basis for \(\mathcal {H}\). In this paper, we study the approximation of skew symmetric operators and provide a \(C^*\)-algebra approach to skew symmetric operators. We classify up to approximate unitary equivalence those skew symmetric operators \(T\in \mathcal {B(H)}\) satisfying \(C^*(T)\cap \mathcal {K(H)}=\{0\}\). This is used to characterize when a unilateral weighted shift with nonzero weights is approximately unitarily equivalent to a skew symmetric operator.