The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-W...The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-Wold Theorem does not hold in the Bergman space. In this paper, using the basis constructed by Michael and Zhu on the Bergman space we prove that each analytic Toeplitz operator M B(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. From the above theorem, we characterize the similarity invariant of some analytic Toeplitz operators by using K 0-group term.展开更多
In this paper,we study the algebra consisting of analytic functions in the Sobolev space W^(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on i...In this paper,we study the algebra consisting of analytic functions in the Sobolev space W^(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.展开更多
In this paper,we study band-dominated operators on Bergman-type spaces and prove that the C*-algebra of band-dominated operators is equal to the essential commutant of Toeplitz operators with a symbol in the set of bo...In this paper,we study band-dominated operators on Bergman-type spaces and prove that the C*-algebra of band-dominated operators is equal to the essential commutant of Toeplitz operators with a symbol in the set of bounded vanishing Lipschitz functions.On the Bergman space and the Fock space,we show that the C*-algebra of band-dominated operators equals the Toeplitz algebra.展开更多
Abstract Let H be a complex seperable Hilbert space and ^（Jt^） denote the collection of bounded linear operators on H. For an operator in L（H）, rad{A}＇ denotes the Jacobson radical of the commutant of A. This pap...Abstract Let H be a complex seperable Hilbert space and ^（Jt^） denote the collection of bounded linear operators on H. For an operator in L（H）, rad{A}＇ denotes the Jacobson radical of the commutant of A. This paper characterizes the similarity of strongly irreducible operator weighted shift in terms of {A}＇/rad{A}＇. Moreover, we suggest some ways to determine when an operator weighted shift is strongly irreducible and when its commutant is commutative.展开更多
It is proved that if is a nonconstant bounded analytic function on the unit ball B n and continuous on S n in C n , and ψ is a bounded measurable function on S n such that T * and T ψ commute, then ψ is the bounda...It is proved that if is a nonconstant bounded analytic function on the unit ball B n and continuous on S n in C n , and ψ is a bounded measurable function on S n such that T * and T ψ commute, then ψ is the boundary value of an analytic function on B n . In addition, the commutants of two Toeplitz operators are also discussed.展开更多
LET B<sub>n</sub> be the unit ball of C<sup>n</sup>. The n-dimensional vector space over the complex field C, S<sub>n</sub>=（?）B<sub>n</sub> is the boundary of B<su...LET B<sub>n</sub> be the unit ball of C<sup>n</sup>. The n-dimensional vector space over the complex field C, S<sub>n</sub>=（?）B<sub>n</sub> is the boundary of B<sub>n</sub>. We use σ to denote the unique rotation-invariant probability measure on S<sub>n</sub>. The Lebesgue spaces L<sub>2</sub>（S<sub>n</sub>, d<sub>σ</sub>） have their customary meaning. H<sup>2</sup>（S<sub>n</sub>）展开更多
The C~*-algebra generated by the strong Θ-operator is considered in this paper. Thepaper contains two parts. The first part shows that the commutant ideal I(T) of C(T) is*isomorphic to where ?_n ??_(n+1), n = 1,2,…a...The C~*-algebra generated by the strong Θ-operator is considered in this paper. Thepaper contains two parts. The first part shows that the commutant ideal I(T) of C(T) is*isomorphic to where ?_n ??_(n+1), n = 1,2,…and ?_n is * isomorphic to the n-matrix algebra over C_0(Z_0). In particular, when T is a completely nonnormal strong Θ-operator, I(T) is * isomorphic to C_0(Z_0)(×)K(I^2). The second part gives the equivalent con-ditions which make the spectrum and the approximate spectrum of the completely nonnormalstrong Θ-operator identify and some K-groups of the C-algebra generated by this classoperator are computed in this part.展开更多
This paper uses the commutant lifting theorem for representations of the nest algebra to deal with the optimal control of infinite dimensional linear time- varying systems. We solve the model matching problem and a ce...This paper uses the commutant lifting theorem for representations of the nest algebra to deal with the optimal control of infinite dimensional linear time- varying systems. We solve the model matching problem and a certain optimal feedback control problem, each of which corresponds with one type of four-block problem. We also obtain a new formula for the optimal performance and prove the existence of an optimal controller.展开更多
This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multi...This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multiplication operators on the Bergman space.展开更多
In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋....In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.展开更多
基金the National Natural Science Foundation of China (Grant No. 10571041)
文摘The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-Wold Theorem does not hold in the Bergman space. In this paper, using the basis constructed by Michael and Zhu on the Bergman space we prove that each analytic Toeplitz operator M B(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. From the above theorem, we characterize the similarity invariant of some analytic Toeplitz operators by using K 0-group term.
文摘In this paper, we characterize the commutant of Toeplitz operators on weighted Bergman space with symbol polynomial by using algebraic curves theory.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10071020).
文摘In this paper,we study the algebra consisting of analytic functions in the Sobolev space W^(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.
基金supported by CSC(201906050022)partially supported by NFSC(11531003)。
文摘In this paper,we study band-dominated operators on Bergman-type spaces and prove that the C*-algebra of band-dominated operators is equal to the essential commutant of Toeplitz operators with a symbol in the set of bounded vanishing Lipschitz functions.On the Bergman space and the Fock space,we show that the C*-algebra of band-dominated operators equals the Toeplitz algebra.
文摘Abstract Let H be a complex seperable Hilbert space and ^（Jt^） denote the collection of bounded linear operators on H. For an operator in L（H）, rad{A}＇ denotes the Jacobson radical of the commutant of A. This paper characterizes the similarity of strongly irreducible operator weighted shift in terms of {A}＇/rad{A}＇. Moreover, we suggest some ways to determine when an operator weighted shift is strongly irreducible and when its commutant is commutative.
基金supported by National Natural Science Foundation of China (Grant Nos.110671041, 10971040)
文摘It is proved that if is a nonconstant bounded analytic function on the unit ball B n and continuous on S n in C n , and ψ is a bounded measurable function on S n such that T * and T ψ commute, then ψ is the boundary value of an analytic function on B n . In addition, the commutants of two Toeplitz operators are also discussed.
文摘LET B<sub>n</sub> be the unit ball of C<sup>n</sup>. The n-dimensional vector space over the complex field C, S<sub>n</sub>=（?）B<sub>n</sub> is the boundary of B<sub>n</sub>. We use σ to denote the unique rotation-invariant probability measure on S<sub>n</sub>. The Lebesgue spaces L<sub>2</sub>（S<sub>n</sub>, d<sub>σ</sub>） have their customary meaning. H<sup>2</sup>（S<sub>n</sub>）
基金Project supported by the National Natural Science Foundation of China.
文摘The C~*-algebra generated by the strong Θ-operator is considered in this paper. Thepaper contains two parts. The first part shows that the commutant ideal I(T) of C(T) is*isomorphic to where ?_n ??_(n+1), n = 1,2,…and ?_n is * isomorphic to the n-matrix algebra over C_0(Z_0). In particular, when T is a completely nonnormal strong Θ-operator, I(T) is * isomorphic to C_0(Z_0)(×)K(I^2). The second part gives the equivalent con-ditions which make the spectrum and the approximate spectrum of the completely nonnormalstrong Θ-operator identify and some K-groups of the C-algebra generated by this classoperator are computed in this part.
文摘This paper uses the commutant lifting theorem for representations of the nest algebra to deal with the optimal control of infinite dimensional linear time- varying systems. We solve the model matching problem and a certain optimal feedback control problem, each of which corresponds with one type of four-block problem. We also obtain a new formula for the optimal performance and prove the existence of an optimal controller.
文摘This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multiplication operators on the Bergman space.
基金the National Natural Science Foundation of China(10571041)the Doctoral Foundation of Hebei Normal University(130144)
文摘In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.
