6 ORIGINS IN SINGLE OPERATOR THEORY the role of the compact operators inif(^f). ThusJ^Jf is a C*-algebra, and we have a natural projection A e ££*- A e «£/Of. In analogy with (1.9), the quotient norm in &/X is given by (1.12) |M||= lim |K||, Ae&, n-* oc where Ax, A2,... is the sequence of truncated matrices Now every *-automorphism of £f is implemented by a unitary matrix U in the sense that 2(A) = UAU~l, A e £. Thus the following definition makes precise in this context the idea that two matrices A and B should have the same asymptotic properties. DEFINITION 1.13. A ~ B iff there is a unitary matrix U such that UAU~l - B e X. What about invariants for the relation A ~ 2?? As before, we have the essential norm defined by (1.12), and the essential spectrum of A e £g, defined as the spectrum of A relative to the C*-algebra &/X. This time, however, neither of these is a complete invariant. Indeed, the situation here is vastly more com- plicated than the corresponding situation for /°°. A new invariant, which has no commutative counterpart, has to do with the Fredholm index. In order to discuss this, let us discard the matrix interpretation and consideri? (resp. X) to be the algebrai^^) of all bounded (resp. compact) operators on a Hilbert space Xoi dimension X0. An operator A is said to be semi-Fredholm if its range AX is closed and one of the two subspaces ker./4 and CokcvA = ker^4* is finite dimensional. In this case the index of A is defined as the number index A = dimker^l - dimker^*. The index of A is an ordinary integer or is ± oo. It is a fact that the semi-Fred- holm operators are stable under compact perturbations, and that the index of such an operator is similarly stable. (1.14) index A = index(^ + K), for every semi-Fredholm operator A and every compact operator K. The formula (1.14) implies that the index is a new invariant for the equivalence relation A - B. Indeed, if A - B and A - XI is semi-Fredholm for some complex scalar A, then B - XI is also semi-Fredholm and moreover (1.15) index(^ - XI) = index(£ - XI). With the help of (1.15), we can now give an example of two inequivalent operators A, B which have the same essential spectrum. Let A be the simple unilateral shift and let B = A*. Then both A and B have the same essential u n,n + l •*tt + l , / 7 + l
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