ON THE SPECTRUM OF POSITIVE OPERATORS by Cheban P. Acharya A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, FL August 2012 Copyright by Cheban P. Acharya 2012 ii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my Ph.D. advisor, Prof. Dr. X. D. Zhang, for his precious guidance and encouragement throughout the research. I am sure it would have not been possible without his help. Besides I would like to thank the faculty and the staffs of the Department of Mathematics who always gave me support by various ways. I will never forget their help during my whole graduate study. And I also would like to thank all members of my thesis's committee. I would like to thank my wife Parbati for her personal support and great pa- tience all the time. My sons (Shishir and Sourav), mother, father, brother, and cousin have given me their support throughout, as always, for which my mere expression of thanks does not suffice. Last, but by no means least, I thank my colleagues in Mathematics department of Florida Atlantic University for their support and encouragement throughout my stay in school. iv ABSTRACT Author: Cheban P. Acharya Title: On The Spectrum of Positive Operators Institution: Florida Atlantic University Dissertation Advisor: Dr. Xiao Dong Zhang Degree: Doctor of Philosophy Year: 2012 It is known that lattice homomorphisms and G-solvable positive operators on Banach lattices have cyclic peripheral spectrum (see [17]). In my thesis I prove that positive contractions whose spectral radius is 1 on Banach lattices with increasing norm have cyclic peripheral point spectrum. I also prove that if the Banach lattice is a KB space satisfying the growth condition and λ is an eigenvalue of a positive contraction T such that jλj = 1, then 1 is also an eigenvalue of T as well as an eigenvalue of T ∗, the dual of T . I also investigate the conditions on contraction operators on Hilbert lattices and AL-spaces which guarantee that 1 is an eigenvalue. As we know from [17], if T : E ! E is a positive ideal irreducible operator on E such that r(T ) = 1 is a pole of the resolvent R(λ, T ), then r(T ) is simple pole with dimN(T − r(T )I) = 1 and σper(T ) is cyclic. Also all points of σper(T ) are simple poles of the resolvent R(λ, T ). Since band irreducibility and σ-order continuity do not imply ideal irreducibility [2], we prove the analogous results for band irreducible, σ-order continuous operators. v DEDICATION My parent, My wife and My sons. ON THE SPECTRUM OF POSITIVE OPERATORS 1 Introduction................................... 1 2 Banach Lattices................................. 3 2.1 Basic Concepts of Banach Lattices...................3 2.2 Sublattices, Solids, Ideals, and Bands..................8 2.3 Positive Operators on Banach Lattices................. 11 2.4 Introduction to Spectral Properties of Positive Operators....... 15 3 Point Spectrum of Positive Contractions................... 22 3.1 Preliminaries............................... 22 3.2 Positive Contractions with 1 as an Eigenvalue............. 25 3.3 The Hilbert Lattice Case......................... 33 4 Spectral Properties of Irreducible, σ-Order Continuous Operators . 38 4.1 Isometries and Eigenvalues........................ 38 4.2 Some Examples.............................. 40 4.3 Basic Results............................... 42 4.4 Main Results............................... 46 Bibliography .................................. 52 vi CHAPTER 1 INTRODUCTION The study of positive operators originated from the study of integral Operators and matrices with nonnegative entries in the beginning of 20th century. However the systematic study of positive operators follows from the study of Riesz Spaces in 1930's by F. Riesz and H. Freudenthal. In the 1940's and 1950's, some of the fundamental work on positive operators was done by Russian mathematicians such as L. V. Kantorovich, M. G. Krein, M. A. Rutman, B.Z. Vulikh and Japanese mathematicians such as H. Nakano, K. Yosida, T. Ogasawara. In 1960's, the study of positive operators speed up. In the last 40 years, the study of positive operators has increased rapidly. In 1974, H. H. Schaefer published his book: Banach Lattices and Positive Operators [17], whose influence on the study of positive operators is enormous. Afterwards many mathematicians have worked and made great contributions in this area. Among those are P.G.Doods, D.H Fremlin, W.A.J Luxemburg, A.C. Zaanen, C. D. Aliprantis, O. Burkinshaw, and P. Meyer Nieberg. The study of spectral properties of positive operators began in early 1960's. It has increased considerably following the publication of H. H. Schaefer's book [17]. More recent books on this topic include [15] and [3]. Spectral theory of both band and ideal irreducible operators is a natural extension of some well-known concepts of irreducible matrices, which were studied by Perron and Frobenious. This study was initiated by And^oand Krienger by investigating spectral 1 radius of some integral operators on function spaces. Later Schaefer, Sawashima, Nirro and De Pagter contributed on the theory of ideal irreducible operators. Over the past few years, Grobler, Schaefer, Casseless, Abramovich, Aliprantis, and Burkinshaw studied band irreducible operators on Banach lattices. In this thesis, we mainly consider eigenvalues of positive operators on Banach lattices. In Chapter 2, we give the basic definitions and theorems that we need in later chapters. All of them can be found in standard text books about positive operators and Banach lattices such as [17],[15], and [3]. In Chapter 3, we focus on the study of eigenvalues of positive contractions on Ba- nach lattices. We prove that a positive contraction on Banach lattices with increasing norm always has cyclic point spectrum. We also prove that if the Banach lattice is a KB-space satisfying the growth condition and if λ is an eigenvalue of a positive contraction T such that jλj = 1. then 1 is an eigenvalue of T as well as an eigenvalue of T ∗, the dual of T . We also investigate the conditions on positive contractions on Hilbert lattices or AL-spaces which guarantee that 1 is an eigenvalue. In chapter 4, we give some examples of isometry operators (actually shift opera- tors) and discuss their spectrum. We give some results about spectral properties of σ-order continuous, band irreducible operators with r(T ) being a pole of the resolvent. We obtain some results that are analogous to the those proved in book [17]. 2 CHAPTER 2 BANACH LATTICES In this chapter we will mention some basic definitions and theorems regarding Banach lattices and positive operators which will be needed for this thesis. All contents on this chapter are well known and can be found in many books, such as [17], [15], and [1]. 2.1 BASIC CONCEPTS OF BANACH LATTICES An ordered set is a nonempty set A endowed with a binary relation ≤, which satisfies the following properties. For all x; y; z 2 A, 1. x ≤ y; y ≤ z ) x ≤ z (transitive property) 2. x ≤ x (reflexive property) 3. x ≤ y; y ≤ x ) x = y (anti-symmetric property) An ordered set (A; ≤) is called a lattice if for each pair (x; y) 2 A×A the element x _ y := supfx; yg and x ^ y := inffx; yg exist in A. A real ordered vector space is a vector space E over R with an ordered relation ≤ which satisfies the following axioms: 1. x ≤ y ) x + z ≤ y + z for all x; y; z 2 E: 2. x ≤ y ) λx ≤ λy for all x; y 2 E and λ ≥ 0. 3 A real lattice ordered vector space is called a Riesz Space or Vector Lattice. Example 2.1. The following are some examples of Riesz spaces. p 1. The vector space l (1 ≤ p < 1) of all real sequences x = (x1; x2;::: ) with P1 p 1 n=1 jxnj < 1, l , the vector space of all real bounded sequences, and c0, the vector space of all real null sequences are Riesz spaces under the usual point-wise ordering. n 2. The Euclidean space R is a Riesz space under the usual ordering (x1; x2; : : : ; xn) ≤ (y1; y2; : : : ; yn) if xk ≤ yk for all k = 1; : : : ; n. 3. Let X be a topological vector space. The vector space C(X) of all real continu- ous functions on X and the vector space Cb(X) of all real bounded continuous functions on X under the ordering f ≤ g if f(x) ≤ g(x) for all x 2 X are Riesz spaces. Let E be a Riesz space. E+ := fx 2 E : x ≥ 0g is called a positive cone of E. A net fxαg in a Riesz space is said to be decreasing ( in symbols xα #) whenever α ≥ β implies xα ≤ xβ. The notion xα # x means that xα # and inffxαg = x both holds. A Riesz space E is called Archimedean if x ≤ 0 holds whenever the set fnx : n 2 Ng is bounded from above. In other words E is called Archimedean x whenever n # 0 holds for each x ≥ 0: For example, order complete Riesz spaces and normed Riesz spaces are always Archimedean. In this thesis all Riesz spaces will be assumed to be Archimedean unless otherwise stated. If x and y are elements of a Riesz space, then the operations (x; y) 7! x _ y, (x; y) 7! x ^ y, x 7! x+, x 7! x− and x 7! jxj are called the lattice operations of the 4 Riesz space, where x+ = x _ 0 x− = (−x) _ 0 jxj = x _ (−x) = x+ + x− x = x+ − x− 0 = x+ ^ x− In above, x+ and x− are called the positive and the negative parts of x and jxj is called the absolute value of x.