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The Essential Self-Commutator of a Subnormal Operator

with John B. Conway, Proc. Amer. Math. Soc. 125 (1997), p. 243-244.
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In this paper an easier proof is obtained of Alexandru Aleman's extension, of an inequality of Axler and Shapiro for subnormal operators, to the essential norm. That is, if S is a subnormal operator and K denotes its essential spectrum, then it is shown that the essential norm of [S*,S] is less than or equal to the square of the distance from z-bar to R(K).

It follows that if S is a subnormal operator, K is its essential spectrum and R(K) = C(K), then [S*,S] is compact.

The method is also applied to show that a hyponormal operator whose essential spectrum has area zero must be essentially normal.