Let X be a real Banach space. A set K ⊆ X
is called a total cone if it is closed
under addition and non-negative scalar multiplication, does not contain both x and
−x for any non-zero x∈X, and is such that
K−K := {x − y : x,
y ∈ K} is dense in X. Suppose that T is a bounded
linear operator on X which leaves a closed total
cone K invariant. We denote by σ(T) and
r(T) the spectrum and 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 an eigenvector in K.
This result was later extended by Nussbaum [6] to
any bounded operator T such that
re(T) < r(T), where
re(T) denotes the essential
spectral radius of T, without the hypothesis of normality. The more general question
of whether r(T) ∈ σ(T) for all bounded operators
T was answered in the negative by Bonsall [1], who as well as
giving counterexamples described a property of K
called the bounded decomposition property, which is sufficient to guarantee that
r(T) ∈ σ(T).
More recently, Toland [8] showed that if X is a separable
Hilbert space and T is
self-adjoint, then r(T) ∈ σ(T), without any extra
hypotheses on K. In this paper we
extend Toland's results to normal operators on Hilbert spaces, removing in passing
the separability hypothesis.