THE SPECTRAL CONTINUITY OF ESSENTIALLY HYPONORMAL OPERATORS

Abstract

If A is a unital Banach algebra, then the spectrum can be viewed as a function � : A ! S, mapping each T 2 A to its spectrum �(T), where S is the set, equipped with the Hausdorff metric, of all compact subsets of C. This paper is concerned with the continuity of the spectrumvia Browder's theorem. It is shown thatis continuous when � is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide. Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of bounded linear operators acting on H. If T ∈ B(H) write σ(T) and σp(T) for the spectrum and the set of eigenvalues of T, respectively. Let S denote the set, equipped with the Hausdorff metric, of all compact subsets of C. If A is a unital Banach algebra, then the spectrum can be viewed as a function σ : A → S, mapping each T ∈ A to its spectrum σ(T). It is known that the function σ is upper semicontinuous and that in noncommutative algebras, σ does have points of discontinuity. J. Newburgh (17) gave the fundamental results on spectral continuity in general Banach algebras. J. Conway and B. Morrel (7) have undertaken a detailed study of spectral continuity in the case where the Banach algebra is B(H). It seems to be interesting and challenging to identify classes C of operators for which σ becomes continuous when restricted to C. The first result of this study is: σ is continuous on the set of normal operators. On the other hand, Newburgh's argument uses the fact that the inverses of normal resolvents are normaloid (cf. see Solution 105 of (10)) and this argument is extended to the set of hyponormal operators because the inverses of hyponormal resolvents are also hyponormal and hence normaloid.