large Fock空间上Toeplitz算子的Fredholm性质

Fredholm Toeplitz Operators on Large Fock Spaces

令ϕ是属于权函数类W,对于n维复空间ℂn上由ϕ诱导的large fock空间Fp(ϕ),记Fp(ϕ)=Lp(ϕ)⋂H(ℂn),其中H(ℂn)是ℂn上的解析函数全体。首先介绍了权函数类W的基本性质、Bergman空间上的覆盖定理并对再生核函数做出了估计。其次引入一个辅助算子Gl,并证明了Gl的有界性、Toeplitz算子Tf的Berezin变换和紧致性之间的关系以及Hankel算子Hf的有界性和紧致性。最后,给定0<p<∞,利用Toeplitz算子和Hankel算子的有界性和紧致性以及large Fock空间Fp(ϕ)的对偶理论,研究了Toeplitz算子Tf在Fp(ϕ)上具有Fredholm性质的充要条件是f的Berezin变换在整个n维复空间上有界且在无穷边界上远离原点,其中f是一个有界的消失平均震荡函数。

Letϕbelongs to the weight functions class W,for a large fock space Fp(ϕ)induced byϕon the n-dimensional complex space,denoted Fp(ϕ)=Lp(ϕ)⋂H(ℂn),where H(ℂn)are the entirety of analytic functions onℂn.Firstly,this paper is the preparation work which mainly introduces the basic properties of weight class W,the covering lemma of Bergman spaces and some estimates of reproducing kernel functions.Secondly,we introduce an auxiliary operator Gl and prove some properties including the boundedness of Gl,the relationship between Berezin transform and compactness of Toeplitz operator Tf and the boundedness and compactness of Hankel operators Hf.Finally,fix 0<p<∞,we use the boundedness and compactness of Toeplitz operator Hf and Hankel operators Tf and the dual theory of large Fock spaces to study the Toeplitz operator Tf is Fredholm on Fp(ϕ)if and only if the Berezin transform of f is bounded away from zero outside a sufficiently large disk in the n-dimensional complex plane,where f is a bounded function of vanishing mean oscillation.