Strongly Irreducible Decompositions Of Operators In Direct Integral Forms And Their Similarity Invariants

In operator theory, people always have interest in similarity invariants of operators. On finite dimensional Hilbert spaces, people characterized a similarity invariant of operators with the Jordan canonical form theorem, meanwhile Jordan matrices are also considered as building blocks to construct operators. This has developed into the reduction theory.On complex separable Hilbert spaces, the inner structure of operators is more compli-cated. To make the inner structure of operators become more clear, people tried to gener-alize the Jordan canonical form theorem in many ways. Different methods lead to different progress. In the introduction of this dissertation, we recall this part of history in detail. We need to point out that in 1978, Zejian Jiang suggested to replace Jordan matrices with strongly irreducible operators to make researches in related topics. At the same time, he constructed an example to show that there exists a bounded linear operator which can not be written as a direct sum of strongly irreducible operators. In the following 30 years, Chunlan Jiang, Chekao Fang, Zongyao Wang, Peiyuan Wu, Youqing Ji, Junsheng Fang, and many other people did lots of work around this class of operators.The main result of this dissertation is strongly influenced by the above researches. We can consider a direct sum of operators as the discrete form of a direct integral of operators. To generalize direct sums of operators to direct integrals of operators, we need to intro-duce measure theory and theory of von Neumann algebras. We first consider whether every bounded linear operator is similar to a direct integral of strongly irreducible operators.In Chapter 1, we construct two examples to show that there are bounded linear operators which are not similar to direct integrals of strongly irreducible operators. Then we prove that a bounded linear operator is similar to a direct integral of strongly irreducible operators if and only if there is a bounded maximal abelian set of idempotents in the commutant of the operator. At the end of this chapter, we show that many n-normal operators are similar to direct integrals of strongly irreducible operators. And the class of operators which are similar to direct integrals of strongly irreducible operators forms a dense subset of L(H) in the operator norm.In Chapter 2, we first show that on countably infinite dimensional Hilbert spaces, the strongly irreducible decomposition of the identity operator I is not unique up to similarity. This is an essential difference between finite dimensional Hilbert spaces and countably infinite dimensional Hilbert spaces. By this property, we can construct bounded linear operators which are not normal and whose strongly irreducible decomposition are not unique up to similarity. In the discussion, we introduce the upper triangular representation for operator-valued matrices. This representation simplifies the proofs in the rest dramatically and makes some facts easier to be observed. We notice that the multiplicity functions of the multiplication operators in the principal diagonal have strong relationship with the strongly irreducible decomposition of the matrix, while the invertibility of the multiplication operators in the superdiagonal has important relationship with the strongly irreducibility of the decomposition of the matrix. In the rest, we show two classes of operators have unique strongly irreducible decompositions, meanwhile we compare the superdiagonals of these matrices and find that when the superdiagonal entries are all invertible multiplication operators, the structure is better.