ABSTRACT.Let /: £{)i) ^Xbea continuous function from the algebra of all bounded linear operators acting on a complex infinite dimensional Hubert space U into a Tr-topological space X. If ¡{WAW'1) = ¡{A) for all A in Z(U) and all invertible W, then / is a constant function.The same result is true for a function / satisfying the above conditions defined on a connected open subset of £{M)o = {T £ £{H): T has no normal eigenvalues}.
Introduction.Let ¿(M) denote the algebra of all (bounded linear) operators acting on a complex Hilbert space U. The classical spectral functions (a (spectrum), sp (spectral radius), oe (Calkin essential spectrum), er;re (Wolf essential spectrum, i.e., the complement in the complex plane C of the semi-Fredholm domain), etc.), mapping Z()i) (endowed with the norm-topology) into the space X of all compact subsets of C (Hausdorff metric), or X = the real interval [0, oo) (the usual topology) have a very erratic behavior.Indeed, if M is infinite dimensional, all these functions, and the uncountably many analyzed in [3], with the single exception of the spectral radius, are continuous on a dense subset of £,(M), and discontinuous on another dense subset of £()i)\The spectral radius, on the other hand, is continuous on an open dense subset of £.(){), but not everywhere. (We can also find certain "natural" spectral functions which are discontinuous everywhere; see the above reference.)Is this behavior a peculiarity of our particular functions? Or, is it possible to construct some "natural" spectral function which is continuous everywhere? The answer is: NO. The existence of discontinuities for all these functions is in the nature of things, not a peculiarity of the special functions considered in the literature, and the deep reason is that the spectral functions are similarity-invariant; that is, they take the same value on all the elements of the similarity orbit
