Closedness of ranges of upper triangular operator matrix
Chunfang Shao · Journal of Shandong University · 2012
Let A∈B(H),B∈B(K).Defined the operator MC by MC=(A C0B),in which C∈B(K,H).Basing on the technique of block operator matrix,the sufficient and necessary conditions of the closedness of R(MC) for every C∈B(K,H) are discussed,when R(A) and R(B) are both closed.Furthermore,the following two questions are researched:(ⅰ) the sufficient and necessary conditions when the range R(MC) is not closed for every C∈B(K,H),when R(A) is not closed,R(B) is closed and the R(A) is closed,but R(B) is not closed;(ⅱ) the sufficient and necessary conditions when the range R(MC) is not closed for every C∈B(K,H),when both R(A) and R(B) are not closed.