Suppose that A and B are bounded linear operators on a Banach space such that AB is a Fredholm operator. In general, BA is not a Fredholm operator. In this note we show that BA is a generalised Fredholm operator in the sense of Caradus. Let X be an infinite-dimensional complex Banach space and let L(X ) denote the Banach algebra of all bounded linear operators on X. By F(X ) we denote the ideal of all finitedimensional operators in L(X ), byL L we denote the quotient algebra L(X )/F(X ) and by

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... e denote the equivalence class of A 2 L(X ) inL L, i.e. =A+F(X ). Moreover, by N(A) and A(X ) we denote the kernel and the range of A, respectively. As usual, A 2 L(X ) is called a Fredholm operator if dim N(A) and codim A(X ) are both finite. It is well known that is Fredholm then A(X ) is closed ; furthermore, A is relatively regular, i.e. there exists a pseudo-inverse S 2 L(X ) such that ASA=A (see [3, · 74]). *