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
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.
