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Hypercyclic Extensions of an Operator on a Hilbert Subspace with Prescribed Behaviors

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Hypercyclic Extensions of an Operator on a Hilbert Subspace with Prescribed Behaviors HYPERCYCLIC EXTENSIONS OF AN OPERATOR ON A HILBERT SUBSPACE WITH PRESCRIBED BEHAVIORS Gokul R Kadel A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Kit Chan, Advisor Rachel Reinhart, Graduate Faculty Representative Juan B`es So-Hsiang Chou ii ABSTRACT Kit Chan, Advisor A continuous linear operator T : X ! X on an infinite dimensional separable topological vector space X is said to be hypercyclic if there is a vector x in X whose orbit under T , orb(T; x) = fT nx : n ≥ 0g = fx; T x; T 2x; : : :g is dense in X. Such a vector x is said to be a hypercyclic vector for T . While the orbit of a hypercyclic vector goes everywhere in the space, there may be other vectors whose orbits are indeed finite. Such a vector is called a periodic point. More precisely, we say a vector x in X is a periodic point for T if T nx = x for some positive integer n depending on x. The operator T is said to be chaotic if T is hypercyclic and has a dense set of periodic points. Let M be a closed subspace of a separable, infinite dimensional Hilbert space H with dim(H=M) = 1. We say that T : H ! H is a chaotic extension of A : M ! M if T is chaotic and T jM = A. In this dissertation, we provide a criterion for the existence of an invertible chaotic extension. Indeed, we show that a bounded linear operator A : M ! M has an invertible chaotic extension T : H ! H if and only if A is bounded below. Motivated by our result, we further show that A : M ! M has a chaotic Fredholm extension T : H ! H if and only if A is left semi-Fredholm. Our further investigation of hypercyclic extension results is on the existence of dual hypercyclic extension. The operator T : H ! H is said to be a dual hypercyclic extension of A : M ! M if T extends A, and both T : H ! H and T ∗ : H ! H are hypercyclic. We actually give a complete characterization of the operator having dual hypercyclic extension on a separable, infinite dimensional Hilbert spaces. We show that a bounded linear operator A : M ! M has a dual hypercyclic extension T : H ! H if and only if its adjoint A∗ : M ! M is hypercyclic. iii To my mother iv ACKNOWLEDGMENTS I would like to express my sincere appreciation to my advisor Dr.Kit Chan for his patience, guidance, and continuous support in preparation of this dissertation. He introduced me to the research area of hypercyclicity and taught me everything I needed to know to become a matured mathematician. I would never have been what I am today without his carefully planned advising. I wish to thank my committee members Dr. Rachael Reinhart, Dr. So-Hsiang Chou and Dr. Juan B`esfor their time and advice. Dr. Chou has always helped me in identifying my potential and grow my confidence level. I would like to thank Dr. Tong Sun for his support as a graduate coordinator. I would like to thank the faculty and staff of the Mathematics and Statistics Department for all of their help, guidance, and advice. Many thanks go to department secretaries Barbara, Marcia, and Mary who were always there to help me in the administrative and academic needs whenever needed. I would also like to thank Bowling Green State University for the financial support, without which I would have never completed my degree. Special thanks go to my friends Leo Pinheiro and Swarup Ghosh. It can never be over- stated how much I learned in company with these talented young mathematicians. Leo has always stood by my side in my good and bad times and has taught me not only mathematics but also the art of living. Lastly, I wish to thank my mother, my sister Subidya Mishra, my brothers Pradhyumna and Shreedhar for all the support. Their constant source of inspiration made me who I am today. Most importantly, I wish to thank my wife Usha for her patience, love, support and encouragement. Without Usha, none of this would be possible. TABLE OF CONTENTS CHAPTER 1: HYPERCYCLICITY AND EXTENSION RESULTS .... 1 1.1 INTRODUCTION . 1 1.2 HYPERCYCLIC OPERATORS . 2 1.3 HYPERCYCLICITY CRITERION . 4 1.4 HYPERCYCLIC VECTORS . 5 1.5 HYPERCYCLIC EXTENSION OF OPERATORS . 6 CHAPTER 2: RESTRICTIONS OF AN INVERTIBLE CHAOTIC OPER- ATOR TO ITS INVARIANT SUBSPACES .................. 7 2.1 INTRODUCTION . 7 2.2 INVERTIBILITY OF A CHAOTIC EXTENSION . 9 2.3 FREDHOLM OPERATORS . 25 2.4 HYPERCYCLIC SUBSPACES . 27 CHAPTER 3: DUAL HYPERCYCLIC EXTENSION FOR AN OPERA- TOR ON A HILBERT SUBSPACE ....................... 29 3.1 INTRODUCTION . 29 3.2 TECHNICAL RESULTS FOR CONSTRUCTING A HYPERCYCLIC VEC- TOR........................................ 31 3.3 MAIN THEOREM ON DUAL HYPERCYCLIC EXTENSION . 61 vi BIBLIOGRAPHY .................................... 71 APPENDIX ........................................ 75 1 CHAPTER 1 HYPERCYCLICITY AND EXTENSION RESULTS 1.1 INTRODUCTION Hypercyclicity is the study of continuous linear operators that possess a dense orbit. Al- though the first examples of hypercyclic operators date back to the first half of the last century, a systematic study of this concept has only been undertaken since the mid-eighties. Seminal papers like the unpublished but widely disseminated thesis of Kitai [21], a highly original and broad investigation by Godefroy and Shapiro [13] and deep operator-theoretic contributions by Herrero [19, 20] were instrumental in creating a flourishing new area of analysis. Definition 1. A continuous linear operator T : X ! X on a separable Fr´echetspace X is said to be hypercyclic if if there is some x 2 X whose orbit under T , orb(T; x):=fT nx : n ≥ 0g = fx; T x; T 2x; : : :g, is dense in X. In such a case, x is called a hypercyclic vector for T . The set of hypercyclic vectors for T is denoted by HC(T ). The origin of this terminology is easily explained. For a long time, operator theorists 2 have been studying cyclic vectors in connection with the invariant subspace problem. Here, a vector x 2 X is said to be cyclic for T if the linear span of its orbit, span fT nx : n ≥ 0g, is dense in X. A vector x 2 X is called supercyclic for T if its projective orbit, fλT nx : n ≥ 0; λ 2 Kg, is dense in X. Operators that possess a cyclic (or supercyclic) vector are called cyclic (or supercyclic, respectively). This suggested the name of hypercyclicity for the case when the orbit itself is dense. The invariant subspace problem, which is open to this day, asks whether every operator on a separable infinite dimensional Hilbert space possesses an invariant closed subspace other than the trivial ones given by f0g and the whole space. Counterexamples do exist for operators on non-reflexive spaces like `1. Enflo [11] offered a negative solution for a specific Banach space, which was later transferred by Read [27] to an operator on `1 that has every non-zero vector hypercyclic. 1.2 HYPERCYCLIC OPERATORS The very first example of hypercyclic operator is given by Birkhoff [4]. In 1929, he proved the existence of an entire function f whose successive translates by a non-zero constant a are arbitrarily close to any function in the space of entire functions H(C). That is, there exists an entire function f and a non-zero constant a 2 C, so that for every g 2 H(C), the sequence f(z + nka) ! g(z) locally uniformly on C, for some sequence fnkg that depends on the function g. Thus, the translation operator Ta has a dense orbit ff(z); f(z +a); f(z +2a);:::g in H(C). MacLane [23], in 1952, gave another example of a hypercyclic operator. He showed that there exists a universal entire function f whose successive derivatives are dense in H(C). Namely, there is an entire function f having the property that for every g 2 H(C) there (nk) exists a sequence fnk % 1g so that f (z) ! g(z) locally uniformly in C, and hence the differentiation operator D on H(C) is hypercyclic. 3 Another early example of hypercyclicity was given by Rolewicz [26] in 1969. This was the first example of hypercyclicity for an operator on a Banach space. The setting is the sequence space `p for 1 ≤ p < 1, and the operator is constructed from the backward shift B defined on `p as follows: B(x1; x2; x3;:::) = (x2; x3;:::); p p p where (x1; x2; x3;:::) 2 ` . B itself is a contraction on ` (kBxk ≤ kxk for each x 2 ` ), so it cannot be hypercyclic. However, Rolewicz proved that the operator T = λB with λ > 1 is hypercyclic. The field of hypercyclicity was actually born in 1982 with the Ph.D. dissertation of Kitai [21] and later with the paper of Gethner and Shapiro [12] and in more than two decades since, it has become the focus of active research by numerous authors. Hypercyclicity is a wide-spread phenomenon in analysis. Perhaps the reason one might expect otherwise lies in the simple realization that finite-dimensional spaces do not support hypercyclic operators. The following result of Kitai [21] clarifies this point. Theorem 2 (Kitai, [21]). There are no hypercyclic operators on finite dimensional spaces. On the other hand, Ansari [1] showed that space with some nice structure always support a hypercyclic operator. Theorem 3 (Ansari, [1]). Every separable, infinite dimensional Fr´echetspace carries a hypercyclic operator. Finally, we mention here a surprising result of Grivaux [14] which confirms a large supply of hypercyclic operators in spaces with nice structures.
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