【摘要】 首先讨论了Dirichlet空间上Toeplitz算子组Fredholm谱的表示 ,证明了 :当 φi∈H∞1 (D) +C1 ( D) (i=1 ,2 ,… ,n)时 ,(Tφ1 ,Tφ2 ,… ,Tφn)的右Fredholm谱SP,re(Tφ1 ,Tφ2 ,… ,Tφn)与Fredholm谱SP,e(Tφ1 ,Tφ2 ,… ,Tφn)相同 ;当 φi∈C1 ( D) (i=1 ,2 ,… ,n)时 ,(Tφ1 ,Tφ2 ,… ,Tφn)的左Fredholm谱SP,le(Tφ1 ,Tφ2 ,… ,Tφn)与Fredholm谱SP,e(Tφ1 ,Tφ2 ,… ,Tφn)相同 .然后讨论了Dirichlet空间上Toeplitz算子与算子组的凸性问题 .证明了乘法算子Mz 是非凸型的 ,这与Hardy ,Bergman空间上所有乘法算子都是凸型算子不同 .也证明了 :T =(Tz,Tz2)不是联合凸型算子 ;若 φi∈H∞1 (D) (i=1 ,2 ,… ,n) ,则W(Tφ1 ,Tφ2 ,… ,Tφn)是凸集 .本文还给出了一个一般性的结论 :假定H为Hilbert空间 ,T∈B(H)为一个有界线性算子 ,当n =2m时有σ(Tm,Tn) ={(λm,λn)λ∈σ(T) }更多还原

【Abstract】 The representation of the Fredholm spectra of the Toeplitz operators on the Diri chlet spaces is discussed first. It is shown that when φ i ∈H ∞ 1(D)+C 1(),i=1,2,…, n , the right Fredholm spectra S P, re (T φ 1 ,T φ 2 ,…,T φ n ) of ( T φ 1 ,T φ 2 ,…,T φ n ) is the same as the Fredholm spectra S P, e (T φ 1 ,T φ 2 ,… ,T φ n ) , and that when φ i∈C 1(), i=1,2,… ,n , the left Fredholm spectra S P, le (T φ 1 ,T φ 2 ,…,T φ n ) is the same as S P, e (T φ 1 ,T φ 2 ,…,T φ n ) . The problem of the convexity o f Toeplitz operators and multi Toeplitz operators on Dirichlet spaces is discus sed next. It is proved that the multiplication operator M z is non convex, which is different from that all multiplication operators on Hardy and Bergman spaces are convex. It is also proved that T=(T z,T z 2 ) is not joint convex operato rs and that if φ i ∈H ∞ 1 (D),i=1,2,…, n , then W(T φ 1 ,T φ 2 ,…,T φ n ) is a convex set. Meanwhile a general result is given: let H be a Hilbert space, T∈B(H) be a bounded linear operator and n=2m , t hen σ(T m ,T n )={(λ m ,λ n )|λ∈ σ(T)} exists.更多还原