University of Alberta EXTENSION OF THE PERRON-FROBENIUS THEOREM TO SEMIGROUPS OF POSITIVE OPERATORS ON ORDER-CONTINUOUS BANACH LATTICES by Aaron Levin A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematical and Statistical Sciences Edmonton, Alberta Spring 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-54726-7 Our file Notre reference ISBN: 978-0-494-54726-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduce, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Nnternet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. 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ABSTRACT In 2000, Heydar Radjavi provided an extension of the Perron-Probenius theorem to semigroups of positive compact operators on Lp spaces. We care­ fully go through Radjavi's proof, rigourously expanding many arguments in an effort to provide a clear and concise treatment. We also extended Rad­ javi's proof to semigroups of positive, compact operators on order-continuous Banach lattices. This extension uses many techniques not used in Radjavi's proof. ACKNOWLEDGEMENT First and foremost I would like to thank my co-supervisor Vladimir Troit- sky for his exceptional abilities as a teacher, supervisor, and friend. I am also indebted to the functional analysis group at the University of Alberta for their generous feedback during the initial stages of this project. In addition, I would like to thank Matthew Mazowita for his scholarly insight and Peter Pivovarov for his tex-nical assistance. Finally, the completion of this thesis would have been impossible without the editorial assistance and emotional support of Marie Leblanc Flanagan. Table of Contents 1 Introduction 1 2 Preliminaries 4 2.1 Banach Spaces 4 2.2 Spectral Theory for Banach Spaces 5 2.3 Banach Lattice Theory 11 3 Decomposable Semigroups 17 4 Positive Projections 22 5 Perron-Frobenius Extension 39 6 Examples and Conclusions 43 Bibliography 47 Chapter 1 Introduction The Invariant Subspace Problem remains one of the most prominent open conjectures in analysis. As of late [11], order structures have been used as an attempt to approach the problem from a new direction. Though activity in the area of positivity has burgeoned only recently, the roots of this approach have been present since the turn of the century. The Perron-Frobenius theo­ rem provides much insight into the possible techniques the invariant subspace problem can be tackled using ordering structures. To introduce the traditional Perron-Frobenius theorem we need two con­ cepts: indecomposability and a partial order on any finite-dimensional space. An n x n matrix is said to be indecomposable if there exist no nontrivial in­ variant norm-closed order-ideals. For an n-dimensional space X there is a naive partial-order accomplished by first picking a basis and ordering coordinate- wise (i.e. x = (xi, • • • , xr) < y = (y\, • • • ,yr) if and only if Xj < yi for all i (after an appropriate basis is chosen)). With this partial order, all norm- closed order-ideals are standard subspaces. The traditional finite-dimensional Perron-Frobenius Theorem for a positive matrix is: Theorem 1.1. [10] Let A be an n x n indecomposable matrix. Then there is a vector x with positive entries, unique up to scalar multiples, such that Ax = r(A)x (where r(A) is the spectral radius). In sum, an indecomposable positive matrix has a largest positive eigenvalue 1 with positive eigenvector. More recently, Heydar Radjavi and Peter Rosenthal extended the finite-dimensional Perron-Probenius theorem to semigroups of positive matrices: Theorem 1.2. [8, Theorem 5.2.6] Let 6 be an indecomposable semigroup of positive matrices on X and denote the minimal positive rank in K+(5 by r. If R+(5 has a unique minimal right ideal, then the following hold: (i) there is a vector x with positive entries, unique up to scalar multiples, such that Sx — r(S)x for all SG6; (ii) every S £ & has at least r eigenvalues of modulus r(S), counting multi­ plicities; these are all of the form r(S)Q with 0r! = 1; (Hi) after a permutation of the basis, & has an r x r block partition such that the block matrix {Sij)lj=1 of each nonzero member S of & has exactly one nonzero block in each block-row and each block-column; (iv) if, for any S 6 6, the block matrix Sij has a cyclic pattern (i.e., there ex­ ists a permutation {ii, • • • ,ir} of {1, • • • ,r} such that the nonzero blocks of & are precisely (Siui2, Si2ti3, • • • , S^^), then o~(S) is invariant under the rotation about the original by the angle ^-; (v) r = 1 if and only if some member of & has at least one positive column. This theorem has been extended from the original Perron-Probenius the­ orem only by the usage of semigroups of positive matrices rather than single matrices. The block partition of the matrix allows for the existence of invariant norm-closed ideals. In 1999 Heydar Radjavi [7] successfully extended the finite-dimensional Perron-Probenius theorem to semigroups of positive compact operators on Lp spaces. The presence of compact operators in Radjavi's extension is somewhat intuitive since they are a very close infinite-dimensional analog to matrices. 2 We know that Lp spaces are a large subclass of order-continuous Banach lattices and therefore the goal of this thesis is an extension of the Perron- Frobenius theorem to order-continuous Banach lattices using Radjavi's ap­ proach. At first glance, Radjavi's theorem seemed to translate with ease to the language of order-continuous Banach lattices. However, along the way many novel deviations from this path were found and a much more clear, con­ cise, and self-contained proof is presented. As well, these results now apply to a much broader class of objects like Orlicz-Lorentz spaces and other interesting order-continuous Banach lattices. 3 Chapter 2 Preliminaries 2.1 Banach Spaces Let us briefly visit some essential concepts in functional analysis that are pertinent to this thesis. For a more thorough dissection of the material please see [4]. A normed space is a pair (X, ||-||) where X is a vector space and ||-|| : X —> K+ is a norm, i.e. for all x, y G X and A G K (where K is E or C): (i) ||x|| = 0 4=>- x = 0 (ii) ||Ax|| = |A|||x|| ("i) ll* + i/ll<IMI + l|y|| The presence of a norm allows us to induce a metric on the vector space X by defining a distance between elements x,y G X as d(x,y) = \\x — y\\. If the induced metric is complete, i.e. all Cauchy sequences have a limit in the vector space, then we call the pair (X, ||-||) a Banach space. We denote the norm-closure of a set A C X by A. If Z, Y are normed spaces then we call an operator T : Z —» Y linear if for all a,p G IK and x,y G Z -we have T (ax + /3y) = aT(x) + (3T(y). A linear operator is bounded if there exists K 3 M > 0 such that for all 4 x G Z \\Tx\\ < M||x||. The space of all linear operators from Z into Y is denoted by C (Z,Y). The space of all linear bounded operators from Z into Y is denoted B(Z,Y). If Y — Z we denote this space by B(Z) for brevity. All operators in this thesis will be assumed linear and bounded. We can turn B (Z, Y) into a normed space by defining the norm of a linear operator ||T|| = sup{||Tx|| | x G X, \\x\\ = 1}. We also say an operator T G B(X) is a projection if T2 = T, which also means that X can be decomposed into X = Range T © ker T. A subspace Y of a vector space X is T-invariant if TY C Y. We denote the restriction of T to any subspace ZClas T\z.