算子方程AX+X*B=C的解

赵琳琳

(德州学院数学系,山东德州 253023)

算子方程AX+X*B=C的解

赵琳琳

(德州学院数学系,山东德州 253023)

利用算子的广义逆及相关投影,研究了一类算子方程的可解性,得到了方程可解的若干条件,并给出了解的一般表示.最后利用算子的矩阵表示,得到了此类算子方程可解的又一充要条件,进而丰富了这方面的研究.

Hilbert空间；算子方程；Moore-Penrose逆

1 引言

令H,K表示任意的Hilbert空间,B(H,K)表示从H到K的所有有界线性算子的集合,记B(H,H)=B(H).对给定的算子A∈B(H,K),B∈B(K,H),C∈B(K),本文研究算子方程

2 主要结果

显然,(8)式有解等价于(12),(13)式有公共解.由引理2.1,得,

命题得证.

[1]Djordjevic D S.Explicit solution of the operator equation A*X+X*A=B[J].J.Comput.Appl.Math., 2007,200:701-704.

[2]Xu Q X,Sheng L J,Gu Y Y.The solutions to some operator equations[J].Linear Algebra Appl.,2008,429:1997-2024.

[3]田学刚,王少英.算子方程AX-XA*=B的正解与实正解[J].纯粹数学与应用数学,2010,26(6):1047-1052.

[4]Piao F X,Zhang Q L,Wang Z F.Thesolution to matrix equation AX+XTC=B[J].Journal of the Franklin Institute,2007,344:1056-1062.

[5]Ben-Israel A,Greville T N E.Generalized Inverses Theory and Applications[M].2nd ed.NewYork:Springer, 2003.

[6]Dajic A,Koliha J J.Positive solution to the equations AX=B and XB=D for Hilbert space operators[J]. J.Math.Anal.Appl.,2007,333:567-576.

Solutions of the operator equation AX+X*B=C

Zhao Linlin

(Department of Mathematics,Dezhou University,Dezhou253023,China)

A class of the operator equations is studied by using the general inverse of the operator and its related projectors,some solvability conditions and a general solution to this equation are obtained.Finally,using the matrix form of the operator,another necessary and sufficient condition for the solvability of such operator equation is derived.These results enrich the research in this area.

Hilbert space,operator equation,Moore-Penrose inverse

O177

A

1008-5513(2012)04-0469-06

2011-08-03.

国家自然科学基金(10971070,11071079).

赵琳琳(1981-),博士,讲师,研究方向：数值代数.

2010 MSC:47A62