Positivity 12 (2008), 185–192
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010185-8, published online October 29, 2007
Triviality of the Peripheral Point Spectrum
of Positive Semigroups on Atomic Banach
Manfred P. H. Wolﬀ
Mathem. Institut d. Universit¨at T¨ubingen in memoriam H. H. Schaefer
Abstract. Generalizing a result of Keicher  we show that generators of
-semigroups on super-atomic Banach lattices have trivial periph-
eral point spectrum provided they satisfy a certain growth condition.
Mathematics Subject Classiﬁcation (2000). 47D06; 47B65; 47A10; 46B42.
Keywords. peripheral point spectrum, positive semigroups, atomic Banach
Let E be an atomic Banach lattice with order continuous norm (for notions not
explained here we refer to [6, 7] concerning Banach lattices and positive operators,
and to  concerning strongly continuous semigroups). In  V. Keicher proved the
following theorem on the eigenvalues of the generator A of a strongly continuous
bounded semigroup T of positive operators T
Let λ be an eigenvalue of A. Then either (λ) < 0 or λ =0.
This theorem generalizes a result of Davies  who had proved it for positive
contractive semigroups on E =
(N), 1 ≤ p<∞.
In this article we want to generalize Keicher’s result in two directions. First
of all we prove it for a more general class of atomic Banach lattices, including the
lattice c of all convergent complex sequences. Secondly we get the result for one-
parameter semigroups satisfying a certain growth condition introduced by Greiner
. In particular our main theorem applies to operators A with compact resolvent,
or whenever the so-called spectral bound s(A) deﬁned later on is a pole of the
resolvent of A.