| Hardy space H2(D),Bergman space La2 and the weighted Bergman spaces La2-(dAα)(αa>-1)over the disk D are classical analytic function spaces,and the shift operator on them,as a simple classical operator,has gone through quite a long research process and has formed a relatively rich and complete theoretical system.Invariant subspace problem is closely related to operator theory on function spaces.The core of the invariant subspace problem can be summed up in the study of the invariant subspaces of shift operator on Bergman space.Therefore,the study of invariant subspaces of shift operators on La2 is a meaningful and complicated problem.The main way to study invariant subspaces is to classify invariant subspaces or to describe all invariant subspaces accurately by some methods.For example,use the method of function theory.The famous Beurling theorem shows that each invariant subspace M of the shift operator on H2(D)has the form M=η2(D),where η is an inner function.Thus,the whole invariant subspaces of the shift operator on H2(D)are accurately characterized by the method of function theory.But the invariant subspaces of the shift operator on La2 are much more complicated.Up to now,there is no method to describe all its invariant subspaces clearly.The most important work in this area is the concept of Beurling type theorem proposed by Beurling and the proof that the Beurling type theorem on La holds.Beurling type theorem is an important property of invariant sub-spaces.Through this property,invariant subspaces can be well classified,thus the Beurling type theorem has become an important research topic on invariant subspaces of operators.The natu-ral extension of H2(D)in the multi-variable version is the Hardy space H2(Dn)over the polydisk,and the invariant subspaces of the shift operator on H2(D)is naturally extended to submodules on H2(Dn).So studying submodules on H2(Dn)is natural and meaningful,but Ahern and Clark has proved that no submodule of finite codimension in H2(Dn)can be of the form ηH2(Dn)(see[1]),However,there are infinitely many submodules of finite codimension.To describe these,Ahem and Clark proved Theorem 1.39,this theorem shows that the description of finite codi-mensional submodules is,at least conceptually,an algebraic matter.We have also get our first glimpse of the interplay between the algebra and analysis.However,this algebraic description is still very abstract,and we know very little about submodules of infinite codimension in H2(Dn).We study submodules of H2(Dn)often starting from simple and concrete submodules and hope to be able to depict them clearly and get general skills.In this paper,we consider the case of Hardy space H2(D2)over the bidisk and Nψ,φ type quotient modules on it,where ψ(z),φ(w)are nonconstant inner functions and Mψ,φ=[(ψ(z)-φ(w))2]be the related submodules generated by(ψ(z)-φ(w))2.In addition,Shimorin has proved that the Beurling type theorem holds in La2 {dAα)with-1<α≤1.Hedenmalm and Zhu have showed that for any α>4 there exists some Ha type submodules on which the shift operator does not possess wandering subspace property.Therefore,the Beurling type theorem fails on La2(dAα)(α>4).Shimorin conjectured that the critical value for the Beurling type theorem on the weighted Bergman spaces La2(dAα)is α=1,but the case of 1<α≤4 is still open.In this paper,we consider the case of a=2 and let B2 be the shift operator on La2(dA2).We consider type quotient modules and La2(dA2)based on the following facts:(i)When ψ(z)=z,ψ(w)=w,the corresponding Nψ,φ type quotient module is N0=H2(D2)(?)[(z-w)2].Let H01=[z-w](?)[(z-w)2],and let Sz be the compression operator on N0.We can prove that H01 is invariant for Sz and Sz:H01→H01 is unitarily equivalent to B2:La(dA2)→La2(dA2).Therefore,we can use the isometric and analytic properties of the shift operator on H2(D)2)to study the Beurling type theorem on La2(dA2),this fact also inspires us to study the Nψ,φtype quotient modules.(ii)In[5],Arveson conjectured that homogenous submodules d-shift module over the unit ball are essentially normal.A refined conjecture was proposed by Douglas[19]in the case of quotient Bergman modules.Recently in[79,80],Wang and Zhao solved the polydisc version of Arveson’s conjecture by proposing a complete criterion to the essential normality of homoge-neous quotient modules.Obviously,if the inner functions ψ(z)and cp(w)take some appropriate forms,Nψ,φ can become the corresponding homogeneous quotient modules.In addition,investi-gation along this line was initiated in[13,21,78].In[21],some special types of quotient Hardy modules such as N0 and[zr-wj]-(i,j∈Z+)were proved essentially normal.In this paper,if ψ(z)=z,we simply write and Nψ,φas and Nφ respectively.We first consider the case of φ(w)∈ H∞(w),and study some basic properties of Nφ,spectral properties of compression operators Sz and Sw,compactness of evaluation operator L(O)|Nφ and essential normality of Sz.We also study the reducibility of compression operators Sz and Sw on N0.Ifψ(z),φp(w)are nonconstant inner functions,we give a complete characterization for the essential normality of Nψ,φ,and we also studies compactness of evaluation operators L(O)|Nφ and R(O)|Nφ,essential spectrum of Sz op Nφ essential normality of compression operators Sz and Sw on Nφ.We have not solved the Beurling type theorem on La2(dA2)but have proved that B2 possesses wandering subspace property on the Ha type submodules of La2(dA2)(which is different from the case of α>4).We have established Corollary 7.1 which encourages us to study the wandering subspace property of B2 on the Ha1,a2...,an type submodules.We hope that Corollary 7.1 is a general phenomenon in the case of Ha1,a2...,an type submodules.Let H2(γ)be the Hilbert space over the bidisk generated by a positive sequence γ={γnm)n,m>0.In this paper,we also prove that the Beurling type theorem holds for the shift operator on H2(γ)with γ={γnm}n,m≥0 satisfying certain series of inequalities.As a corollary,we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk.Finally,we study the wandering subspace property of the fridge operators on the Mφ type submodules over the bidisk,and some preliminary results are obtained.The paper is organized as follows:In the first Chapter,we will introduce the background of operator theory in function spaces,preliminary knowledge,development status and research ideas of the issues we are concerned about.In Chapter 2,We study some properties of Nφ type quotient modules over the bidisk and the related operators.In Chapter 3,We study some properties of Nψ,φ type quotient modules over the bidisk and the related operators,where ψ(z),φp(w)are both nonconstant inner functions.In Chapter 4,We study the Beurling type theorem on the Hilbert space generated by a positive sequence.In Chapter 5,We study the wandering subspace property of the shift operator on a class of invariant subspaces of the weighted Bergman space La2(dA2).In Chapter 6,We study the wandering subspace property of the fringe operators on the Mφ-type submodules over the bidisk.In Chapter 7,The summarize and prospect... |
