Recently R. G. Douglas showed [4] that if F is a nonunitary isometry and U is a unitary operator (both acting on a complex, separable, infinité dimensional Hilbert space 34?), then V -K is unitarily equivalent to V 0 U (acting on 3rf? ®34? ) where K is a compact operator of arbitrarily small norm. In this note we shall prove a much more general theorem which seems to indicate "why" Douglas' theorem holds (and which yields Douglas' theorem as a corollary). Our theorem is based on the Calkin

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... ra analogue of the following well-known fact: If X is an eigenvalue for the operator T which lies in the boundary of the numerical range of T, then the eigenspace determined by X reduces T. If T and S are operators acting on Hilbert spaces 3tif\ and J^2 respectively, we shall write T ~ 5 if for each e > 0 there is a compact operator K such that T -K is unitarily equivalent to S, and the norm of K is < e. We shall show that a large class of operators have the property that T = T 0 N where N is any normal operator such that <r(N) the spectrum of N lies in a certain set (determined by T). In particular, if T v is a Toeplitz operator and N is a normal operator such that <J(N) lies in the set of extreme points of the convex hull of *(Ty) then T v = T 9 © N. In what follows & (34? ) will denote the algebra of bounded linear operators (henceforth, simply "operators") acting on a fixed complex, separable, infinite dimensional Hilbert space Jf 7 . The Calkin algebra ^ is the C*-algebra which results from forming the quotient space: 3) (ffl ) modulo the ideal of compact operators. For an operator T we shall let T e denote the coset in *$ which contains T. Recall that the spectrum of T is by definition the set a(T) = {\:T -XI is not invertible} and the numerical range of T is by definition the set W(T) = {(Tf, f ) :/ is a unit vector in 34? }. The analogous objects for *$ are the essential spectrum <r e (T) = {\:T e -\I e is not invertible in ^\ and the essential numerical range W e (T) = {p e