Abstract
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 s(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) ¿ s(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) ¿ s(T).
More recently, Toland [8] showed that if X is a separable Hilbert space and T is self-adjoint, then r(T) ¿ s(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.
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) ¿ s(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) ¿ s(T).
More recently, Toland [8] showed that if X is a separable Hilbert space and T is self-adjoint, then r(T) ¿ s(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.
| Original language | English |
|---|---|
| Pages (from-to) | 574-576 |
| Number of pages | 3 |
| Journal | Bulletin of the london mathematical society |
| Volume | 31 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - 5 Sept 1999 |
Keywords
- Analysis