The thesis deals with the spectral behavior of positive and related operators on Banach lattices. We first examine the spectral properties of those positive operators that satisfy the so-called condition (c). As a consequence of this result, we derive some theorems about the behavior of the peripheral spectrum of positive operators satisfying condition (c).
The study of the spectral properties of positive operators began much later and closely followed the development of the theory of normed Riesz spaces (Banach function spaces or Banach lattices). Although there were some earlier extensions to infinite dimensions of the classical Perron-Frobenius theorems (see [SHl]) for positive matrices (matrices with non-negative values), a systematic study was not started until the early years of this year of the spectral theory of positive operators. 1960s. For example, the study of ergodic properties of positive operators is one of the important topics in ergodic theory (see [A], [Nl] and [N2]), and the theory of semigroups of positive operators, which was developed ten years ago. was drafted. , has found many applications in differential equation theory and probability theory (see [DA], [Nl], [N2] and [OS]).
The central problem in the study of spectral behavior of positive operators is to find out how the positivity interacts with other properties of the operators. Another way to investigate the spectral properties of positive operators is to decompose the operators into relatively simpler parts.
Banach Lattices and Positive Operators
Recall that a real Banach lattice E is a real Banach space and a Riesz space at the same time, such that Ixl ::; Iyl in E implies that Ilxll ::; Ilyli. Moreover, the dual Banach lattice E' of any Banach lattice E has a Fatou norm, and every C(X) space has Fatou norm (see [SRI]). If we define a new norm II· Ilr on Lr(E) by IITllr = IIITIII, then Lr(E) becomes a Banach lattice.
Finally, let us quote a result that will be needed later in the study of the spectral properties of positive operators. To study the cyclicity properties of the spectrum of positive operators, we need the following known results. In this section we introduce the center of a Banach lattice and give some important properties of the center without proof.
In this section, we present the sign operator associated with a fixed element in a complete Banach lattice. The sign operator will be needed later when studying the cyclicity of the peripheral spectrum of positive operators.
Spectral Theory of Positive Operators
So there exists a neighborhood of x, say U(x), such that this inequality holds for all points in this neighborhood. By Lemma 2.2.6a there exists a point qo E X such that Tnlxol(qo) = Zo - 1 for all nonnegative integers n. Then there exists a sequence {an} of nonnegative numbers such that Tn ~ anI and limn_=( an)l/n = r(T).
We already see that if T is a positive operator such that its spectrum is properly contained in the unit circle, then T satisfies condition (c), and thus by Theorem 2.2.1 the projection of some power of T into the middle Z(E) ) dominates a positive multiple of the identity operator and thus a non-singular element in the middle. If this were not the case, we could find a positive integer n such that ~(Tn). It therefore follows from the compactness of X that there exists a point Po E X such that Po E Xn for all n.
From this fact we can deduce that there is some positive integer k such that a(Tk) = {I} and then the result follows from Theorem 2.4.1. Now we ask such a question: Let T be a positive contraction on a Banach lattice E such that its spectrum is properly contained in the unit circle {z : Izl = I}. Huijsmans and Ben de Pagter asked the following question: Let T be a positive operator on a Banach lattice E such that a(T) = {I}.
This is the case when T is a lattice homomorphism such that O'(T) = {I} or when T-l is a power bounded operator.
Decomposition Theorems
Then T is a disjunction preserving operator if and only if there exists a map
1fT is a discontinuity-preserving operator on a Banach lattice E such that T satisfies condition (c), then there exists a positive integer k such that Tk E Z(E). We will need this result in Section 4 for the study of reducibility of disconnection-preserving operators. In this section we will show that if T is an order continuous discontinuity preserving operator on a Banach lattice having the Fatou norm and if its spectrum is contained in a sector of angle less than 7r, then T can to be decomposed into the sum of its central part. and its almost powerless part.
Theorem 3.1.6 shows that if an operator T that maintains the disjunction satisfies condition (c), then there will be some power of T in the center Z(E). Now we would like to know under what circumstances an operator that maintains the disjunction will itself be in the center. In addition, every C(X) space has the Fatou norm. Decomposition theorem J) Let T be an order-continuous disjunction-preserving operator on an order-complete Banach lattice E with Fatou norm.
In the following lemmas we will assume that T is an order-conserving operator of continuous discontinuity in a complete Banach network E. On the other hand, since E' has the Fa-tou norm and since T' is a conservation operator of the continuous disconnection of such order. that u(T') = u(T), applying theorem 3.2.4 to T' we see that T' = G + D me. Let T be a discontinuity-preserving operator on an arbitrary Banach lattice F such that (i) T is invertible and (ii) u(T) is contained in a sector 6. i) From Proposition 3.1.5 it follows that T' is an invertible disconnection-preserving operator on F', which has the Fatou norm.
If (j(T) is contained in a sector ~ with an angle less than Jr, then any of the following conditions implies that (jo(T) = (j(T). ii) T' is also a disjointness-preserving operator . In this section, we prove another type of decomposition theorems for disjointness-preserving operators without any restrictions on the spectrum of operators. Let T be an order connected disjointness-preserving operator on E such that T' is also a disjointness-preserving operator .
To do this, we recall that a disjointness-preserving operator T is quasi-invertible on E if the following conditions are satisfied (see [AH]):. i) T is serially connected and T is one-one (injective);. iii) T' is a disjointness-preserving operator. In this work, we assume that E is order-complete and that T is an order-connected operator that preserves disjointness on E.
