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
For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) ...For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.展开更多
Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and ...Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and K be bounded linear operators with T<sub>1</sub>KS<sub>1</sub>+T<sub>2</sub>KS<sub>2</sub>=0. Then （1）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K||≥||K||and （2）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K<sub>1</sub>|<sub>2</sub><sup>2</sup>=||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub> ||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||<sub>2</sub><sup>2</sup>=||TX-XS||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if TK+KS;and （4）||TXS-X+K||<sub>2</sub><sup>2</sup>=||TXS-X||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup> if TKS=K. These generalize the results of （1）and （3）.展开更多
Let H K be a pair of Hilbert spaces and i be inclusion from H into K If i is acontraction, then we denote H K In this case, P=ii~* is a positive operator in theHilbert space K,0≤P≤I, and P=i~*i positive operator in ...Let H K be a pair of Hilbert spaces and i be inclusion from H into K If i is acontraction, then we denote H K In this case, P=ii~* is a positive operator in theHilbert space K,0≤P≤I, and P=i~*i positive operator in the Hilbert space H, 0≤P≤I and0∈σ_p(P). L. de Branges proved that in this case there is a unique complement space L ofH in K such that L K and for x∈H, v∈L,it holds that (x+y,x+y)_k≤(x,x)_H+ (y,y)_LAnd for:z∈K there is unique decomposition z=x+y for which the above equality holds.展开更多
We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples...We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal.By relating the weak k-hyponormality and k-hyponormality of a commuting operator pair to positivity of restriction of some linear functionals to corresponding cones of functions,we prove that there is an operator pair that is polynomially hyponormal but not 2-hyponormal,generalizing Curto and Putinar’s result(1991,1993)to the two-variable case.展开更多
In this paper, necessary and sufficient conditions are obtained for unilateral weighted shifts to be near subnormal. As an application of the main results, many answers to the Hilbert space problem 160 are presented a...In this paper, necessary and sufficient conditions are obtained for unilateral weighted shifts to be near subnormal. As an application of the main results, many answers to the Hilbert space problem 160 are presented at the end of the paper.展开更多
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This work was supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars,the Ministry of Education of China
文摘For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.
文摘Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and K be bounded linear operators with T<sub>1</sub>KS<sub>1</sub>+T<sub>2</sub>KS<sub>2</sub>=0. Then （1）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K||≥||K||and （2）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K<sub>1</sub>|<sub>2</sub><sup>2</sup>=||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub> ||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||<sub>2</sub><sup>2</sup>=||TX-XS||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if TK+KS;and （4）||TXS-X+K||<sub>2</sub><sup>2</sup>=||TXS-X||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup> if TKS=K. These generalize the results of （1）and （3）.
基金Project supported by the National Natural Science Foundation of China.
文摘Let H K be a pair of Hilbert spaces and i be inclusion from H into K If i is acontraction, then we denote H K In this case, P=ii~* is a positive operator in theHilbert space K,0≤P≤I, and P=i~*i positive operator in the Hilbert space H, 0≤P≤I and0∈σ_p(P). L. de Branges proved that in this case there is a unique complement space L ofH in K such that L K and for x∈H, v∈L,it holds that (x+y,x+y)_k≤(x,x)_H+ (y,y)_LAnd for:z∈K there is unique decomposition z=x+y for which the above equality holds.
基金supported by National Natural Science Foundation of China(GrantNos.10801028 and 11271075)Science and Technology Development Planning Program of Jilin Province(GrantNo.201215008)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20120043120003)
文摘We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal.By relating the weak k-hyponormality and k-hyponormality of a commuting operator pair to positivity of restriction of some linear functionals to corresponding cones of functions,we prove that there is an operator pair that is polynomially hyponormal but not 2-hyponormal,generalizing Curto and Putinar’s result(1991,1993)to the two-variable case.
文摘In this paper, necessary and sufficient conditions are obtained for unilateral weighted shifts to be near subnormal. As an application of the main results, many answers to the Hilbert space problem 160 are presented at the end of the paper.
