Perturbations of the spectra of upper triangular operator matrices
Xiaohong Cao · Journal of Shaanxi Normal University · 2009
The Kato lower semi-Fredholm spectrum of an upper triangular operator matrix on a Hilbert space is discussed.By means of the relationship between n(T) and d(T) of two operators on the diagonal of an upper triangular operator matrix,some sufficient conditions for an upper triangular operator matrix to be a Kato lower semi-Fredholm operator are given.It is proved that if B is a Kato lower semi-Fredholm operator and n(B)=∞,then MC=AC 0B is a Kato lower semi-Fredholm operator for some operator C.Meanwhile,the perturbation of the Kato lower semi-Fredholm spectrum of an upper triangular operator matrix is discussed.It is proved that if for any λ∈σ(B),B*-λI is a Saphar operator and d(B*-λI)=∞,then ∩C∈ B(K,H)σlk(MC)=σlk(B)∪{λ∈C:A-λI is compact}∪(σlk(A)∩ρ(B))=σSF-(B)∪{λ∈C: A-λI is compact}∪{σlk(A)∩ρ(B)}.