Fock正交补空间上亚正规对偶Toeplitz算子

Hyponormal dual Toeplitz operators on the orthogonal complement of the Fock space

Fock空间是由整函数构成的一类重要解析函数空间.由于与量子力学中的海森堡群表示理论相关,所以不仅在函数空间算子理论有着重要研究意义,还与量子力学的研究紧密相关.对偶Toeplitz算子也是算子理论研究内容之一,且与Toeplitz算子相互影响.算子的亚正规性,起源于Halmos的Hilbert空间问题集中的第5个问题,一直都是算子理论的重点研究内容之一.主要利用函数空间的结构和对偶Toeplitz算子的性质,刻画Fock空间的正交补空间上以(z-a)(z-b)为符号的对偶Toeplitz算子亚正规性,并得出其亚正规性等价于正规性,同时也探究了以调和多项式为符号的对偶Toeplitz算子是亚正规的一些必要条件.

Fock space is an important analytic function space composed of entire functions.Because it is related to the representation theory of Heisenberg group in quantum mechanics,Fock space has important research significance not only in the operator theory of function space,but also closely related to the study of quantum mechanics.Dual Toeplitz operator is also one of the research contents in operator theory,interacting with Toeplitz operator.The hyponormality of operators,which originated from the fifth problem in Halmon’s book named“A Hilbert Space Problem”,has always been one of the key research contents in operator theory.In this paper,we use the structure of function space and the properties of dual Toeplitz operators to characterize the hyponormality of dual Toeplitz operator with symbol(z-a)(z-b)on the orthogonal complement of Fock space,and obtain that its hyponormality is equivalent to normality.We also explore some necessary conditions for the dual Toeplitz operator with the harmonic polynomial symbol to be hyponormal.