| A big trend for the development of modern mathematics is to get various branchesconnecting with each other. We will consider the chaotic properties of linear systemcoming from the natural intersection between topological dynamical system and operatortheory in the frst part of this thesis. We will emphasize the mutual infuence of classicthought, notions and conclusions in these theories in order to accelerate the developmentof them.On the one hand, many problems in the physical world （for example, the N bodyproblem） tell us that determinism is not enough for us to solve real problems. We alsoneed to consider randomness and disorder of our world. That is the beginning of chaostheory. On the other hand, one big task in operator theory is trying to understand thestructure of operators. The most famous problem relating to it in operator theory is theinvariant subspace problem. The research of it cause the development of hypercyclic op-erators. In fact, the hypercyclicity is the same thing to transitivity in dynamical system.Up to now, there are many breakthrough progresses in the theory of hypercyclicity: Kitai,etc. gave a criterion for hypercyclicity, Herrero gave the spectral picture description of theclosure of the set of all hypercyclic operators on Hilbert space, Gethner, Shapiro, Salasgave the description of hypercyclic weighted shift operators, Grosse-Erdmann generalized the thought of them and considered the same problem on Frechet space, Costakis andManoussos generalized the hypercyclicity by borrowing a defnition from topological dy-namic system, they defned the J class operators and got related theories, Chan provedthat the closed span of all hypercyclic operators on a separable, infnitely Hilbert spaceH is L（H）. Based on the junction here, people came to go one step further and beganto study the chaotic properties of linear operators. Now, the theory is ongoing. Herrerogot that there exist many Devaney chaotic operators in L（H）, Grosse-Erdmann gave thedescription of Devaney chaotic weighted shift operators, Hou Bingzhe, Liao Gongfu, CaoYang, Bermudez, Bonilla, Martinez-Gimenez and Peris considered the Li-Yorke chaoticproperties for weighted shift operators and gave a sufcient condition for being Li-Yorkechaotic, In2010s, Hou Bingzhe, Cui Buyu and Cao Yang considered the distribution-al chaoticity of Cowen-Douglas operators and gave a criterion for being distributionallychaotic, called norm-unimodal.In the frst part, we will consider the distributionally chaotic operators and Li-Yorkechaotic operators from the global point of view. Specifcally,Chapter One. we give a brief survey on the background studied in this dissertation.Some preliminary knowledge in dynamical system and operator theory is reviewed, whichwill be used in later chapters.Chapter Two. First we begin with particular classes of operators and obtain thatcompact operators and normal operators can not admit any chaos （distributional chaosand Li-Yorke chaos）. We also review the chaotic properties for weighted shift operatorsand Cowen-Douglas operators. Second we describe the closures and interiors of the setsof all distributionally chaotic operators and Li-Yorke chaotic operators in the languageof spectral picture. It turns out that the closures and interiors are all the same thoughdistributional chaos is stronger than Li-Yorke chaos in defnition. We also compare theclasses of norm-unimodal operators and distributionally chaotic operators and get that thechaotic operators which are invariant under small perturbations must be norm-unimodaloperators. Third we prove that the closures and interiors are all arc connected. Last wegive the description of the closure of the set of all J class operators in the language ofspectral picture which Costakis and Manoussos propose to consider in their paper.As for structure of operators, we can think of it in another way. In the matrixtheory of fnite dimensional space, the famous Jordan Standard Theorem sufciently reveals the internal structure of matrices. Jordan Theorem indicates that the eigenvaluesand the generalized eigenspaces of matrix determine the complete similarity invariantsof a matrix. Any matrix can be written in the direct sum of Jordan blocks uniquely inthe sense of similarity. If we regard Jordan block as brick, we can build up any “matrix”building using these bricks. When we consider a complex, separable, infnite dimensionalHilbert space H, we face one of the most fundamental problems in operator theory,that is how to build up a theorem in L（H） which is similar to the Jordan StandardTheorem in matrix theory, or how to determine the complete similarity invariants ofthe operators. The complexity of infnite dimensional space makes it impossible to fndgenerally similarity invariants. The main difculty behind this is that it is impossible forpeople to fnd a fundamental element in L（H）, similar to Jordan’s block, so as to constructa perfect representation theorem. It is because of the introduction of the concept ofirreducible operators by Halmos in1968that Voiculescu obtained the well-known Non-commutative Weyl-von Neumann Theorem for general C*-algebra. But irreducibility isonly a unitary invariant and can not reveal the general internal structure of operatoralgebras and non-self-adjoint operators. Since the1970s, some mathematicians haveshowed their concern for the problem on Hilbert space operator structure in two aspects.In one aspect, the mathematicians such as Foias, Ringrose, Arveson, Davidson etc. havemade great eforts to study the structures of diferent classes of operators or operatoralgebras, such as Toeplitz operators, weighted shift operators, quasinilpotent operators,triangular and quasitriangular operators, triangular and quasitriangular algebras etc.In the other aspect, they have set up the approximate similarity invariants for generaloperators by introducing the index theory and spectral picture as tools. One of the mosttypical achievements, made by Apostol, Filkow, Herrero and Voiculescu is the theoremof similarity orbit of operators. This theorem suggests that the fne spectral picture isthe complete similarity invariant as far as the closure of similarity orbit of operatorsare concerned. Besides, in the1970s, Gilfeather and Jiang, Z.J. proposed the notion ofstrongly irreducible operator respectively. And Jiang, Z.J. frst thought that the stronglyirreducible operators could be viewed as the suitable replacement of Jordan block inL（H）. An operator will be considered strongly irreducible if its commutant contains nonontrivial idempotent. In the theory of matrix, strongly irreducible operator is Jordanblock up to similarity. Through more than20years’ research, Jiang Chunlan etc. have shown that strongly irreducible operators are suitable replacement of Jordan blocks inL（H） defnitely and constructed the Jordan Theorem. The signifcance is profound.In the second part, i.e. in Chapter Three, we will construct the polar decompositiontheorem by strongly irreducible operators. The classical polar decomposition theoremtell us that any operator T can be written as the multiplication of a partial isometryand a positive operator, i.e. T=U|T|or U T=|T|, where U is a partial isometryand|T|=（T*T）1/2. We are going to substitute|T|by a strongly irreducible operator.Specifcally,Theorem3.2.1. For any T∈L（H） and any∈>0, there exist a partial isometryU, a compact operator K with||K||<∈and a strongly irreducible operator S such thatT=（U+K）S. |
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