Functions of unitary operators: Derivatives and trace formulas

Abstract

Let A be a bounded self-adjoint operator on a separable Hilbert space H and U 0 an arbitrary unitary operator. We derive formulas for higher order derivatives of the function s → φ ( e i s A U 0 ) for sufficiently smooth functions φ . We compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of Krein–Koplienko–Neidhardt.