Properties Of Compressed Shifts On The Hardy Space Of The Bidisk

The operator theory on function spaces is an important and popular branch of functional analysis,and the famous work is introduced by Beurling.The main idea of operator theory on function spaces is to study the problem of the operator theory(such as the invariant subspace)by the tools of complex analysis,harmonic analysis and so on.The motivation of this paper comes from Nagy-Foias’s operator model theory.We main study the reducibility,the spectrum and invariant subspaces of compressed shifts in this paper.The paper is organized as follows:In the first chapter,we introduce background of the operator theory on function spaces,and some notions of function spaces and operators,such as the Hardy space,the vector-valued Hardy space,the unilateral shift,the compressed shift and so on.In Chapter 2,we give a necessary and sufficient condition for the reducibility of a Co(2)operator by using its characteristic function.Moreover,we obtain the number of reducing sub-spaces of a C0(2)operator.Later,we give some examples of Co(2)operators.In Chapter 3,we study the compressed shift Sz1 on the Beurling type quotient module Kθof Hardy space/H2(D2)over the bidisk.Firstly,we give a necessary and sufficient condition such that Sz1 has nontrivial pure isometry reducing subspaee.As an application,we show that Sz1 has Agler reducing subspaces if and only if θ=φ(z1)Ψ(22)is the product of two one-variable inner functions.In Chapter 4,we study the reducibility of the compressed shift Sz1 on the Beurling type quotient module K0 over the bidisk.Firstly,for a rational inner function θ with degree(n,1),we show that Sz1 is reducible on K0 if and only if Sz1 has Agler reducing subspaces.Furthermore,we study the rational inner function with degree(n,2),and this case is qciite different from the degree of(n,1).In Chapter 5,we study the spectrum of compressed shifts,and we give a complete charac-terization for the spectrum of compressed shifts on Kθ when θ is a rational inner function on D2 with degree(1,1).In Chapter 6,we give a characterization for some special invariant subspaces of compressed shifts on Kθ when θ is a rational inner function on D2 with degree(1,1).