Peripheral Spectra of Order-Preserving Normal Operators

Let X be a real Banach space. A set K X is called a total coneif it is closed under addition and non-negative scalar multiplication,does not contain both x and –x for any non-zero xX, andis such that K–K:= {x–y:x, yK} is dense in X. Supposethat T is a bounded linear operator on X which leaves a closedtotal cone K invariant. We denote by (T) and r(T) the spectrumand spectral radius of T. Krein and Rutman [5] showed that if T is compact, r(T) >0 and K is normal (that is, inf{||x + y||: x, y K, ||x|| =||y|| = 1} > 0), then r(T) is an eigenvalue of T with aneigenvector in K. This result was later extended by Nussbaum[6] to any bounded operator T such that r_e(T)<r(T), wherer_e(T) denotes the essential spectral radius of T, without thehypothesis of normality. The more general question of whetherr(T) (T) for all bounded operators T was answered in the negativeby Bonsall [1], who as well as giving counterexamples describeda property of K called the bounded decomposition property, whichis sufficient to guarantee that r(T) (T). More recently, Toland [8] showed that if X is a separable Hilbertspace and T is self-adjoint, then r(T) (T), without any extrahypotheses on K. In this paper we extend Toland's results tonormal operators on Hilbert spaces, removing in passing theseparability hypothesis. 1991 Mathematics Subject Classification47B65.