. FA ] 5 S ep 1 99 7 A NOTE ON PAIRS OF PROJECTIONS

We give a brief proof of a recent result of Avron, Seiler and Simon. In [1], it is proved that if P, Q are (not necessarily self-adjoint) projections on a Hilbert space and (P − Q) n is trace-class (i.e. nuclear) for some odd integer n then tr (P − Q) n is an integer and in fact, if P and Q are self-adjoint, tr (P − Q) n = dim E 10 − dim E 01 where E ab = {x : P x = ax, Qx = bx}; (see also [2]). The proof given in [1] uses the structure of the spectrum of P − Q and Lidskii's theorem; it is therefore not applicable to more general Banach spaces. The purpose of this note is to give a very brief proof of the same result which involves only simple algebraic identities and is valid in any Banach space with a well-defined trace (i.e. with the approximation property). We use [A, B] to denote the commutator AB − BA. The basic material about operators on Banach spaces which we use can be found in the book of Pietsch [3]. We summarize the two most important ingredients. We will need the following basic result from Fredholm theory. Suppose X is a Banach space and A : X → X is an operator such that for some m, A m is compact. Let S = I − A; then F = ∪ k≥1 S −k (0) is finite-dimensional and if Y = ∩ k≥1 S k (X) then Y is closed and X can be decomposed as a direct sum X = F ⊕Y. Furthermore F and Y are invariant for S and S is invertible on Y. We refer to [3] 3.2.9 (p. 141-142) for a slightly more general result. We will also need the following properties of nuclear operators and the trace. If X is a Banach space then an operator T : X → X is called nuclear if it can written as a series T = ∞ n=1 A n where each A n has rank one and ∞ n=1 A n < ∞. The nuclear operators form an ideal in the space of bounded operators. When X has the approximation property, one can then define the trace of T unambiguously by tr T = ∞ n=1 tr A n (where the trace of a rank one …