Agler Decomposition And Modular Space Of Matrix-valued Rational Functions On Bidisc

Letθbe an inner function on the Hardy space H²（D） and let （?） be its model space.The associated compressions of the shift （?） played a pivotal role in both operator and function theory.Indeed,allowingθto be operator valued,the famous Sz-Nagy Foias model theory says:every completely nonunitary operator C₀contraction is unitarily equivalent to a compression of the shift S_θ on a model space for θ is a operator-valued inner function.On Hardy space H²（D）,Beurling’s theorem found all S _z-shift invariant subspaces in H²（D）.But on bidisc,there must be more complicated.Agler shows that every function of?in H₁^∞associates a reproducing Hilbert function space （? ）on D².If θ is the function of H²（D²）,it is equivalent to model space H_θ.We study the bidisc within the matrix-valued rational functionθ（z）and the corresponding model space （?）.Generalize Moore’s theorem on vector-valued analytic functions space,and give a complete description thatH_θhas a unique operator-valued regen-erating kernel.We can write the structure of a Hilbert space when （?） is reproducing Kernel function of it.In Chapter 2,we consider the Agler decomposition of inner functions on D²,and the u-niqueness.In Chapter 3,we give the generalization of vector case for matrix-valued rational function θ（z）on bidisc.We define associated model spaceH_θⁱand_θⁱ.According to the rela-tion betweenH_θⁱand_θⁱ,we describe the struction ofH_θⁱandH_θ,and study the uniqueness of their Agler decompositions.We provide the model spaces of two matrix-valued rational func-tions θ₀,θ₁,and give the conclusion of uniqueness of the Agler decomposition.In Chapter 4,generalize the properties of scalar-valued reproducing Kernel.The relation between vector-valued reproducing kernels are studied through the spaces.According to the conclusions of operator-valued regenerating kernels and theorems in scalar-valued case,we give the sufficient and necessary conditions that₁,₂are reducing subspaces of compressed shift operators S_z1,S_z2on H_θ.Finally,we can prove that S₁,S₂are reducing subspaces of com-pressed shift operators S_z1,S_z2.