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Let XC be a separable, infinite dimensional, complex Hilbert space, and let E(9C) denote the algebra of all bounded linear operators on SC. In [2] it was shown that every quasinilpotent operator T in E(jC) (i.e., every T whose spectrum is the singleton {O}) is a norm-limit of nilpotent operators. The purpose of this note is to give a very short and easily remembered proof of this theorem. Our proof has two basic ingredients: the use of a model for quasinilpotent operators constructed in [4], and the use of a deep theorem of Voiculescu [7]. For completeness we review the relevant facts from [4] and [7] that we shall need. Let X. = X ED X e ... be the direct sum of S0 copies of SC indexed by the positive integers, let K = K,,' I= be a monotone decreasing sequence of nonnegative numbers converging to zero, and let UK denote the quasinilpotent backward weighted shift operator in Ef(f3C) defined by the equation UK(X19 X2, * * * 9 Xn9 .. . ) = (K1X2, K2X3,. . .9 KnXn, 19 . .. ). (1) THEOREM A [4]. If T is any quasinilpotent operator in e(9C), then there is a sequence K = K(T) = {Kn,) 'I with the above described properties and a subspace 91 of 'K.C that is invariant under the quasinilpotent operator UK such that T is similar to UK191 and such that the subspace 'CK e )1 is infinite dimensional.

That quasinilpotent operators are norm-limits of nilpotent operators revisited