On the peripheral spectrum of positive operators

Abstract

This paper contributes to the analysis of the peripheral (point) spectrum of positive linear operators on Banach lattices. We show that, under appropriate growth and regularity conditions, the peripheral point spectrum of a positive operator is cyclic and that the corresponding eigenspaces fulfil a certain dimension estimate. A couple of examples demonstrates that some of our theorems are optimal. Our results on the peripheral point spectrum are then used to prove a sufficient condition for the peripheral spectrum of a positive operator to be cyclic; this generalizes theorems of Lotz and Scheffold.

Appendix: The signum operator

In this appendix we shortly recall some facts about the signum operator on complex Banach lattices which are needed in the article. First we recall the following result from [3, Section C-I.8]:

Proposition A.1

Let E be a complex Banach lattice and let $f \in E {\setminus } \{0\}$. Then there exists a unique linear operator $S_f$ on $E_{|f|}$ which fulfils the following two conditions:

• (a) $S_f \overline{f} = |f|$, where $\overline{f}$ denotes the complex conjugate vector of f.

• (b) $|S_fg| \le |g|$ for all $g \in E_{|f|}$.

The operator $S_f$ is called the signum operator associated to $E_f$. If we identify $E_{|f|}$ with a $C(K;\mathbb {C})$-space (K compact) by means of the Kakutani representation theorem such that |f| corresponds to the constant 1-function on K, then we have $S_{f}g = fg$ for each $g \in C(K;\mathbb {C})$ where the multiplication is computed in $C(K;\mathbb {C})$. Now we come to the major definition in this appendix (compare [3, Definitions B-III.2.2(b) and C-III.2.1]).

Definition A.2

Let E be a complex Banach lattice and let $f \in E {\setminus } \{0\}$. By means of the Kakutani representation theorem we can identify the principal ideal $E_{|f|}$ with a $C(K;\mathbb {C})$-space for some compact Hausdorff-space K such that |f| corresponds to the constant 1-function on K. Using the multiplication on $C(K;\mathbb {C})$, we define $f^{[n]} := f^n$ for each $n \in \mathbb {Z}$.

Note that $f^{[n]} = S_f^n |f|$ whenever $n \ge 0$ and $f^{[n]} = S_{\overline{f}}^n |f|$ whenever $n < 0$. This shows that the definition of $f^{[n]}$ is independent of the choice of the representation $E \xrightarrow {\sim } C(K;\mathbb {C})$. The following property of $f^{[n]}$ is important:

Proposition A.3

Let E be a complex Banach lattice and let $f \in E {\setminus } \{0\}$. Suppose that $h \in E_+$ with $f \in E_h$ and identify the principal ideal $E_h$ with a $C(K;\mathbb {C})$-space for some compact Hausdorff space K, where h corresponds to the constant 1-function on K. In the space $C(K; \mathbb {C})$ the vectors $f^{[n]}$ are given by

$$\begin{aligned} f^{[n]}(x)= {\left\{ \begin{array}{ll} (\frac{f(x)}{|f(x)|})^n |f(x)| &{} \quad \text {if } f(x) \not = 0 \\ 0 &{} \quad \text {if } f(x) = 0 \text {.} \end{array}\right. } \end{aligned}$$

(1)

Proof

First, let $n \in \mathbb {N}_0$. Define an operator $\tilde{S}_{f}$ on $E_{|f|} = C(K;\mathbb {C})_{|f|}$ which is given by

$$\begin{aligned} \tilde{S}_{f} g(x) = {\left\{ \begin{array}{ll} \frac{f(x)}{|f(x)|} g(x) &{}\quad \text {if } f(x) \not = 0 \\ 0 &{}\quad \text {if } f(x) = 0 \end{array}\right. } \end{aligned}$$

for every $g \in C(K;\mathbb {C})_{|f|}$ and every $x \in K$. Then $\tilde{S}_f$ fulfils properties (a) and (b) from Proposition A.1 and we thus have $\tilde{S}_f = S_f$. This implies $f^{[n]} = S_f^n |f| = \tilde{S}_f^n |f|$, which proves the assertion. For $n < 0$ one argues similarly, using the operators $\tilde{S}_{\overline{f}}$ and $S_{\overline{f}}$ instead. $\square$

We can now prove the following lemma which is a slight modification of [3, Lemma C-III.3.11].

Lemma A.4

Let E be a complex Banach lattice and let $G,H \subset E$ be to vector subspaces of E. Let $n \in \mathbb {Z}$ and assume that $f^{[n]} \in H$ for each $f \in G {\setminus } \{0\}$. Then $\dim G \le \dim H$.

Proof

The proof is very similar to the proof of [3, Lemma C-III.3.11]; for the convenience of the reader, we include it here. Let $0 < m \le \dim G$, let $g_1,\ldots ,g_m$ be linearly independent elements of G and define $u := |g_1| + \cdots + |g_m|$. Then we can identify the principal ideal $E_u$ with a $C( K; \mathbb {C})$-space. There are points $x_1,\ldots ,x_m \in K$ and functions $f_1,\ldots ,f_m \in C(K;\mathbb {C})$ with the same linear span as $g_1,\ldots ,g_m$ which have the property that $f_j(x_k) = \delta _{jk}$ (where $\delta _{jk}$ is the Dirac delta) for all $j,k \in \{1,\ldots ,m\}$; this can easily be seen by an induction over m. By our assumption, we have $f_j^{[n]} \in H$ for all $j \in \{1,\ldots ,m\}$ and due to Proposition A.3 we can compute $f_j^{[n]}$ in $C(K;\mathbb {C})$ by means of formula (1). Hence, we also have $f_j^{[n]}(x_k) = \delta _{jk}$ for all $j,k \in \{1,\ldots ,m\}$ and therefore, the vectors $f_1^{[n]},\ldots ,f_m^{[n]}$ are linearly independent. Thus $m \le \dim H$, which proves the assertion. $\square$