Left(Right) Browder Spectra of 2*2 Upper Triangular Operator Matrices
Huaijie Zhong · Journal of Huaqiao University · 2007
Let X and Y be complex Banach spaces,Mc be an upper triangular operator matrix with given A∈B(X),B∈B(Y) and any C∈B(YX),σlbMc={λ∈C:Mc)-λB+(XY)} be the left Brooder spectra and σrb(Mc)={λ∈C: Mc)-λB-(XY)} be the right Browder spectra of Mc,where B-(XY)={T∈Φ-(XY): des(T)∞}.It is shown that the passage from σlb(Mc)(σrb(Mc)) to σlb(A)∪σlb(B)|σrb(A)∪σrb(B) is accomplished by filling holes,that is,there is an equality σ*(A)∪σ*(B)=σ*(Mc)∪W,where W is the union of certain holes in σ*(Mc) and σ*∈{σlb,σrb}.Furthermore,the exact location of W is found.