【作者】 王春梅；

【导师】 纪友清；

【作者基本信息】 吉林大学 ， 基础数学， 2005， 硕士

【摘要】 本文首先考虑H~2(T)空间上以n阶Blaschke乘积为符号的Toeplitz算子酉等价于(?)T_z,并由此推出这类算子的有限强不可约分解在相似下唯一。然后证明了H~p(T)(1≤p≤∞,p≠2)空间上以n阶Blaschke乘积为符号的Toeplitz算子相似于(?)T_z,从而推出这类算子的有限BIR分解在相似下唯一。更多还原

【Abstract】 In this paper, T is the unit circle in the complex plane C, and L^p（T）（1 ≤ P ≤ +∞） denotes the linear space consists of the p-th integrable functions with respect to the normalized arc lenghth measure dθ/2π L^∞（T） denotes the space made up of the essentially bounded measurable functions on T. HP（T） is the classic Hardy space , and {zⁿ}_n=0^∞ is the base, where z = eⁱθ.For f∈L^∞（T） the Toeplitz operator T_f on HP（T） （1 < p < +∞） is defined by Tfg = Pfg for g in HP（T）, where P is the projection of L^p（T） onto H^P（T） and P is the orthogonal projection when p=2. Tf is called an analytic Toeplitz operator when f ∈H^∞（T）. But the definition is valid when p=l or 00. The reason is that there isn’t a bounded projection of L¹ （T） （L^∞（T）） onto H¹（T） （H^∞（T））. But we may consider the problem from the angle that the mutipliction operator resticts its invariant subspace. For φ ∈L^∞（T） the niu-tiplication operator Mφ on L^P（T） is defmed by Mφf = φf for f in L^p（T）. Because H¹（H^∞（T）） is an invariant subspace of Mφ on L¹（T） （L^∞（T）） when φ in H^∞（T）, Mφ|H¹（T） （Mφ|H^∞（T）） is an operator on H¹（T） （H^∞（T））,which is also called a Toeplitz operator on H¹（T） （H^∞（T）） and denoted by Tφ.Toeplitz operators not only have an important relation with other subjects, for example, physics, probability theory and information control theory and so on, but also are the important no-selfadjoint operators except for the differential operators. Hence Toeplitz operators play a large role in operator theory.In operator theory it is very important that operators are classed according to similarity. In a finite dimentional space two matrixs are similar if andonly if they have the same Jordan canonical form. However, in an infinite dimentional space there isn’t the analogue of Jordan block , and there isn’t the Jordan’s canonical form therem, either. To characterize the similarity of operators, it is an important means to look for the approximate Jordan’s canonical form therem. Professor Jiang Zejian think that the BIR. operator is a suitable analogue of Jordan block and the general bounded operator may be constructed with them. Later, the seminar of Jilin University and their co-workers built the decomposable therem of structure of operators in an infinite dimentional space and obtained the approximate canonical therein in an infinite dimentional space. Hence, it is very significative to consider similarity of Toeplitz operators, especially to consider whether an operator is similar to the direct sum of some BIR operators. Such problem is considered in this paper. We begin with a trival result on H2（T）.nTherem 2.1 q{z) = J| *^"J is a Blaschke product of n order on T, where |<x,| < 1 and eij denotes the complex conjugate ofctj. TH andTz are the Toeplitz operator on #2（T）. Then Tq^@Tz.Corollary 2.2 The Toeplitz operators Tv on H2 has a unique finitely strongly irreducible decomposition with respect to similarity.From the above therein, we can easily consider that whether there is the same result for Tq on Hp（l） when p ^ 2. Of course, unitary equvilence is impossible, but whether is Tq @TZ right? In order to answer the question, we still define H? = {zjf ° q] f € H1’}, where {e^}"=1 is the orthonormal basis of kerT* which occurs in the proof of therein 2.1. It is obvious that Hv- is a closed subspace of Hp.We can obtain the following propositionproposition 3.1 Tq\Hv Tz when p ^ 2 and 1 < p < -t-oo.更多还原