上三角算子矩阵值域的闭性

Closedness of ranges of upper triangular operator matrix

设A∈B(H),B∈B(K),定义MC=(A C0B),其中C∈B(K,H)。基于算子分块的技巧,讨论了当R(A),R(B)都是闭的时候,对每一C∈B(K,H),R(MC)是闭的充要条件。进而研究了:(ⅰ)当R(A)不闭,R(B)闭时,以及当R(A)闭,R(B)不闭时,对任意C∈B(K,H),R(MC)不闭的充要条件;(ⅱ)当R(A),R(B)同时不闭时,对任意C∈B(K,H),R(MC)不闭的充要条件。

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.