【摘要】 本文讨论可分的复希氏空间上的（有界）算子的Banach约化问题。指出了某些特殊算子的Hilbert约化与Banach约化的等价性,刻划了一类Banach不可约的解析Toeplitz算子的超不变子空间格,讨论了当φ∈H^∞是弱^*生成元时T_φ的循环向量的存在性。本文的主要结果是:于φ∈H∞,T_∞生成的弱闭代数U_Tφ等于T_φ的换位{T_φ}′的充要条件是T_φ的不变子空间皆超不变;这一条件又等价于φ是H^∞的弱⁺⁺生成元。更多还原

【Abstract】 In this paper we consider the Banach reduction of operators acting on a separable complex Hilbert space. An invariant subspace M for operator A is called Banach reducing subspace for A if it has an invariant complement N. It is shown that the condition that M, N be complementary invariant subspaces for A implies that N=M under one of the following hypotheses: 1) A is cyclic normal; 2) A is the direct sum of a finite number of cyctic normal operators with mutually singular spectral measures; 3) A is totally subnormal; i.e., all the operators in {A}′are subnormal, where {A}′denotes the commutant of A.The main object of the paper is to study Banach irreducible （BIR） analytic Toeplitz operators. The analytic Toeplitz operators T_ψ’s are such ones whenever {T_φ}′= {T_z}′. It is proved that a sufficient and necessary condition that {T_φ}′be equal to {T_z}′is that the equality Lat {T_φ}′=Lat T_z holds, where Lat is the invariant subspace lattice for a family of operators. It is also shown that every invertible element f in Hardy space H^∞ is cyclic vector of analytic Toeplitz operator T_φ whenever φ is a weak generator of H^∞. The main result of this paper states: if φ in H^∞ then the following conditions are equivalent: 1) φ is a weak^* generator of H^∞; 2) the weak closed algebra T_φ containing identity generated by T_φ, coincides with the commutant {T_φ}′; 3) the hyperlattice Lat {T_φ}′of T_φ, is equal to the lattice Lat T_φ of T_φ.更多还原