Unitary Classification And Similarity Classification Of N-Tuples Cowen-Douglas Operators

Let H be a complex separable Hilbert space, (?)(H) denote the collection of bounded linear operators on H, (?)~n(H) denote the collection of all commuting ntuplesbounded linear operators on H andΩbe a bounded domain in C~n. We also denote A_m(Ω) of all n-tuples CD operators onΩwith index m, which possess trivial holomorphic bundles. In this paper, by the tools of reproducing kernel theory and functions of several complex variables, we show that each A∈A_m(Ω) possesses spanning holomorphic cross-section ifΩis a multi-cylinder in C~n. Based on the notion of spanning holomorphic cross-section, we obtain a sufficient and necessary conditions of unitary equivalence and similarity equivalence of n-tuples CD operators.Furthermore, we give unitary classification and similarity classification of n-tuples CD operators.This paper contains four parts. In part 1, we introduce the relative background on this paper; In part 2, we give some the necessary preparation knowledges; In part 3, we prove the main results of this paper; In part 4, we conclude the main results and we make some prospects for future works.