Corrigendum to: Operator monotone functions and Löwner functions of several variables

Abstract

We fix a gap in the proof of Theorem 7.24 in Ann. of Math. 176 (2012), 1783–1826. There is a gap in the proof of Theorem 7.24 in [1], though the statement of the theorem is correct. In the proof of necessity, we argue that Λ is in G by contradiction. If it were not, invoking the Hahn-Banach separation theorem would yield a real skew-symmetric matrix K and a constant δ ≥ 0 such that tr(ΓK) ≥ −δ for all Γ in G, and tr(ΛK) < −δ. In the proof we assumed that δ = 0, but this assumption is unjustified. Instead, we argue as follows. Define ∆ by ∆ij = (x r j − xi )Kji, i 6= j, and with the diagonal entries ∆ii chosen so that each ∆ r ≥ 0 and so that