关于算子方程AXB－X＝C的解─—纪念C．Apostol

On the Solution of the Operator Equation AXB-X=C──Dedicated to the memory of Constantin Apostol

本文给出了算子方程ＡＸＢ－Ｘ＝Ｃ可解的若干充要条件，其中（Ａ，Ｂ）为下列情形之一：Ａ或Ｂ有闭值域；Ａ（Ｂ＊）有闭值域并且是单射或者相似于一个协亚正规算子并且Ｂ（Ａ＊）是单侧移位；Ａ＋（Ｂ（＊＋））幂有界，其值域Ｒ（Ａ＋） Ｒ（Ａ）（Ｒ（Ｂ） Ｒ（Ｂ＋）并且Ｂ（Ａ＊）是单侧移位；Ａ＝Ｕ＊且Ｂ＝Ｕ是Ｈａｒｄｙ空间上重数为１的单侧移位．而且，给出了解的表达式．

We give several necessary and sufficient conditions for the solvability of the operator equation AXB-X=C,where(A,B) is one of the following forms:A or B has closed range;A(B*) has closed range and is injective or similar to a cohyponomal operator and B(A*) is a unilateral shift;A+(B(*+)) is power bounded with R(A+) R(A)(R(B) R(B+)) and B(A*) is a unilateral shift;A = U* and B = U the unilateral shift of multiplicity 1 on the Hardy space.Moreover,the explicit expressions of solutions are given.