In this paper we consider the block Toeplitz operators T_Φ on the weighted Bergman space A_α~2(D,C^n) and we give a necessary and sufficient condition for the hyponormality of block Toeplitz operators with symbol in...In this paper we consider the block Toeplitz operators T_Φ on the weighted Bergman space A_α~2(D,C^n) and we give a necessary and sufficient condition for the hyponormality of block Toeplitz operators with symbol in the class of functions Φ=F+G* with matrix-valued polynomial functions F and G with degree 2.展开更多
In this paper,we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space.First we show that the dual Toeplitz operator with the bounded symbol is hyponormal...In this paper,we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space.First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal.Then we obtain a necessary and sufficient condition for the dual Toeplitz operator S_(φ) with the symbol φ(z)=az^(n1zm1)+bz^(n2zm2)(n1,n2,m1,m2∈N and a,b∈C)to be hyponormal.Finally,we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.展开更多
For a hyponormal operator T if there is a polynomial p(·) such that σ(p(T))= {0}, then p(T) =0. In general, it is proved in this way: from σ(p(T)) =0, we know that σ(T) consists of finite points, and therefore...For a hyponormal operator T if there is a polynomial p(·) such that σ(p(T))= {0}, then p(T) =0. In general, it is proved in this way: from σ(p(T)) =0, we know that σ(T) consists of finite points, and therefore T must be normal and so is p(T), hence p(T) =0.展开更多
This paper presents a discussion on a problem suggested by N.Salinas（J.Operator Theory,Vol.10,1983）,i.e.,if T=（T<sub>1</sub>,T<sub>2</sub>,…,T<sub>n</sub>）is a commuting n-tupl...This paper presents a discussion on a problem suggested by N.Salinas（J.Operator Theory,Vol.10,1983）,i.e.,if T=（T<sub>1</sub>,T<sub>2</sub>,…,T<sub>n</sub>）is a commuting n-tuple ofhyponormal operators on a complex Hilbert space,do we have（1）（2）δ（T-μ）=dist（μ,σ<sub>l</sub>（T））,μ∈C<sup>n</sup>?It is proved that the problems（1）and（2）are both true for some commuting n-tuples ofsemihyponormal operators.In general case,problem（1）has a negative answer.As an ap-plication,we point out that there exists a commuting n-tuple of hyponormal operatorsT（n≥2）,such that|λ|【,where σ<sub>T</sub>（T）denotes J.L.Taylor’sjoint spectra of T,even if the condition of Taylor’s joint spectra of T is given.展开更多
In this paper,we show that the hyponormal Toeplitz operator T with trigonometric polynomial symbol is either normal or completely non-normal.Moreover,if T is non-normal,then Tˉ has a dense set of cyclic vect...In this paper,we show that the hyponormal Toeplitz operator T with trigonometric polynomial symbol is either normal or completely non-normal.Moreover,if T is non-normal,then Tˉ has a dense set of cyclic vectors.Some general conditions are also considered.展开更多
基金supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(2009-0093827)
文摘In this paper we consider the block Toeplitz operators T_Φ on the weighted Bergman space A_α~2(D,C^n) and we give a necessary and sufficient condition for the hyponormality of block Toeplitz operators with symbol in the class of functions Φ=F+G* with matrix-valued polynomial functions F and G with degree 2.
基金Partially supported by NSFC(Grant No.11701052)the second author was partially supported by the Fundamental Research Funds for the Central Universities(Grant Nos.2020CDJQY-A039 and 2020CDJ-LHSS-003)。
文摘In this paper,we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space.First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal.Then we obtain a necessary and sufficient condition for the dual Toeplitz operator S_(φ) with the symbol φ(z)=az^(n1zm1)+bz^(n2zm2)(n1,n2,m1,m2∈N and a,b∈C)to be hyponormal.Finally,we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.
基金Project supported by the National Natural Science Foundation of China
文摘For a hyponormal operator T if there is a polynomial p(·) such that σ(p(T))= {0}, then p(T) =0. In general, it is proved in this way: from σ(p(T)) =0, we know that σ(T) consists of finite points, and therefore T must be normal and so is p(T), hence p(T) =0.
文摘This paper presents a discussion on a problem suggested by N.Salinas（J.Operator Theory,Vol.10,1983）,i.e.,if T=（T<sub>1</sub>,T<sub>2</sub>,…,T<sub>n</sub>）is a commuting n-tuple ofhyponormal operators on a complex Hilbert space,do we have（1）（2）δ（T-μ）=dist（μ,σ<sub>l</sub>（T））,μ∈C<sup>n</sup>?It is proved that the problems（1）and（2）are both true for some commuting n-tuples ofsemihyponormal operators.In general case,problem（1）has a negative answer.As an ap-plication,we point out that there exists a commuting n-tuple of hyponormal operatorsT（n≥2）,such that|λ|【,where σ<sub>T</sub>（T）denotes J.L.Taylor’sjoint spectra of T,even if the condition of Taylor’s joint spectra of T is given.
基金Supported by the National Natural Science Foundation of China (Grant Nos.1097102010671028)
文摘In this paper,we show that the hyponormal Toeplitz operator T with trigonometric polynomial symbol is either normal or completely non-normal.Moreover,if T is non-normal,then Tˉ has a dense set of cyclic vectors.Some general conditions are also considered.
