Lambert showed in 1970 that two quasisimilar injective unilateral weighted shifts must be similar and hence have the same dosed range points. But whether the conclusion is true for injective bilateral weighted shift o...Lambert showed in 1970 that two quasisimilar injective unilateral weighted shifts must be similar and hence have the same dosed range points. But whether the conclusion is true for injective bilateral weighted shift operators has not been proved yet. In this note we answer the question affirmatively with a stronger result. We prove that two quasisimilar injective bilat-展开更多
An operator T is called k-quasi-*-A(n) operator, if T^(*k)|T^(1+n)|^(2/(1+n))T^k ≥T^(*k)|T~* |~2T^k , k ∈ Z, which is a generalization of quasi-*-A(n) operator. In this paper we prove some properties of k-quasi-*-A(...An operator T is called k-quasi-*-A(n) operator, if T^(*k)|T^(1+n)|^(2/(1+n))T^k ≥T^(*k)|T~* |~2T^k , k ∈ Z, which is a generalization of quasi-*-A(n) operator. In this paper we prove some properties of k-quasi-*-A(n) operator, such as, if T is a k-quasi-*-A(n) operator and N(T )■N(T~* ), then its point spectrum and joint point spectrum are identical. Using these results, we also prove that if T is a k-quasi-*-A(n) operator and N(T )■N(T ), then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.展开更多
In this article, we give an operator transform T （＊） from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ...In this article, we give an operator transform T （＊） from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ（T ） = σ（ T （＊）） and σa（T ）／{0} = σa（ T （＊））／{0}, in case T belongs to class A. Next, we obtain some relations between T and T （9）.展开更多
We introduce a new family of classes of operators termed as *p-paranormal operator, classes *A(p,p);p > 0 and *A(p,q);p, q > 0, parallel to p-paranormal operator and classes A(p,p);p> 0 and A(p,q);p, q > 0...We introduce a new family of classes of operators termed as *p-paranormal operator, classes *A(p,p);p > 0 and *A(p,q);p, q > 0, parallel to p-paranormal operator and classes A(p,p);p> 0 and A(p,q);p, q > 0 introduced by M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto [1]. We present a necessary and sufficient condition for p-hyponormal operator T∈B(H)to be *p-paranormal and the monotonicity of *A(p,q). We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].展开更多
This paper shows that every operator which is quasisimilar to strongly irreducible Cowen-Douglas operators is still strongly irreducible. This result answers a question posted by Davidson and Herrero (ref. [1]).
Let A and B be quasisimilar operators. We describe refinedly the intersection relationsof the components of various essential spectra of A with various subsets of the essentialspectrum of B, and give an affirmative an...Let A and B be quasisimilar operators. We describe refinedly the intersection relationsof the components of various essential spectra of A with various subsets of the essentialspectrum of B, and give an affirmative answer to a question posed by L. A. Fialkow.展开更多
Yang Liming showed in 1988 that if S is a subnormal operator, T is a hyponormal operator and T and S are quasisimilar, then σ_e(S)(?) σ_e(T), and hence he deduced the conclusion that two quasisimilar subnormal opera...Yang Liming showed in 1988 that if S is a subnormal operator, T is a hyponormal operator and T and S are quasisimilar, then σ_e(S)(?) σ_e(T), and hence he deduced the conclusion that two quasisimilar subnormal operators have equal essential spectra. This is an important result in the theory of quasisimilarity. In this note we improve Yang’s method to show that if S or S is subnormal, T is a bounded linear operator and T and S are quasisimilar, then σ_e(S) (?) σ_e(T).展开更多
Let X denote an infinite dimensional complex Banach space and L(X) denote the set of all the bounded linear operators on X. For A∈L(X), B∈L (Y), A and B are quasisimilar (written as A (?) B) if there exist P:X→Y, Q...Let X denote an infinite dimensional complex Banach space and L(X) denote the set of all the bounded linear operators on X. For A∈L(X), B∈L (Y), A and B are quasisimilar (written as A (?) B) if there exist P:X→Y, Q: Y→X, where P and Q are bounded linear, injective and dense-ranged such that PA=BP, QB=AQ. T. B.展开更多
This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some prop...This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.展开更多
B. Sz-Nagy and C. Foias introduced the concept of quasisimilarity in 1967. Since then, the quasisimilarity of operators has become an important research problem in the theory of operators and a lot of attractive resul...B. Sz-Nagy and C. Foias introduced the concept of quasisimilarity in 1967. Since then, the quasisimilarity of operators has become an important research problem in the theory of operators and a lot of attractive results have been obtained. But, as regards intersection relations between some subsets of the spectrum σ(A)and those of the spectrum σ(B), where A and B are quasisimilar operators, comparatively few results have been delivered.展开更多
文摘Lambert showed in 1970 that two quasisimilar injective unilateral weighted shifts must be similar and hence have the same dosed range points. But whether the conclusion is true for injective bilateral weighted shift operators has not been proved yet. In this note we answer the question affirmatively with a stronger result. We prove that two quasisimilar injective bilat-
基金Supported by the Natural Science Foundation of the Department of Education of Henan Province（12B110025, 102300410012）
文摘An operator T is called k-quasi-*-A(n) operator, if T^(*k)|T^(1+n)|^(2/(1+n))T^k ≥T^(*k)|T~* |~2T^k , k ∈ Z, which is a generalization of quasi-*-A(n) operator. In this paper we prove some properties of k-quasi-*-A(n) operator, such as, if T is a k-quasi-*-A(n) operator and N(T )■N(T~* ), then its point spectrum and joint point spectrum are identical. Using these results, we also prove that if T is a k-quasi-*-A(n) operator and N(T )■N(T ), then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.
基金supported by Science Foundation of Ministry of Education of China (208081)Technology and pioneering project in Henan Provice (102300410012)Education Foundation of Henan Province (2007110016, 2008B110006)
文摘In this article, we give an operator transform T （＊） from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ（T ） = σ（ T （＊）） and σa（T ）／{0} = σa（ T （＊））／{0}, in case T belongs to class A. Next, we obtain some relations between T and T （9）.
文摘We introduce a new family of classes of operators termed as *p-paranormal operator, classes *A(p,p);p > 0 and *A(p,q);p, q > 0, parallel to p-paranormal operator and classes A(p,p);p> 0 and A(p,q);p, q > 0 introduced by M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto [1]. We present a necessary and sufficient condition for p-hyponormal operator T∈B(H)to be *p-paranormal and the monotonicity of *A(p,q). We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].
基金This work was supported by the 973 Project of China and the National Natural Science Foundation of China (Grant No. 19631070).
文摘This paper shows that every operator which is quasisimilar to strongly irreducible Cowen-Douglas operators is still strongly irreducible. This result answers a question posted by Davidson and Herrero (ref. [1]).
基金Project supported by a grant from the Fujian Province Natural Science Foundation.
文摘Let A and B be quasisimilar operators. We describe refinedly the intersection relationsof the components of various essential spectra of A with various subsets of the essentialspectrum of B, and give an affirmative answer to a question posed by L. A. Fialkow.
基金Project supported by the Science Foundation of Fujian Province
文摘Yang Liming showed in 1988 that if S is a subnormal operator, T is a hyponormal operator and T and S are quasisimilar, then σ_e(S)(?) σ_e(T), and hence he deduced the conclusion that two quasisimilar subnormal operators have equal essential spectra. This is an important result in the theory of quasisimilarity. In this note we improve Yang’s method to show that if S or S is subnormal, T is a bounded linear operator and T and S are quasisimilar, then σ_e(S) (?) σ_e(T).
基金Project supported by the Science Foundation of Fujian Province.
文摘Let X denote an infinite dimensional complex Banach space and L(X) denote the set of all the bounded linear operators on X. For A∈L(X), B∈L (Y), A and B are quasisimilar (written as A (?) B) if there exist P:X→Y, Q: Y→X, where P and Q are bounded linear, injective and dense-ranged such that PA=BP, QB=AQ. T. B.
基金Supported by National Natural Science Foundation of China(Grant Nos.10926173,11171066 and 10771034)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.2010350311001)Natural Science Foundation of Fujian Province of China(Grant No.2009J05002)
文摘This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.
文摘B. Sz-Nagy and C. Foias introduced the concept of quasisimilarity in 1967. Since then, the quasisimilarity of operators has become an important research problem in the theory of operators and a lot of attractive results have been obtained. But, as regards intersection relations between some subsets of the spectrum σ(A)and those of the spectrum σ(B), where A and B are quasisimilar operators, comparatively few results have been delivered.
