算子方程AX+X*B=C的解

趙琳琳

(德州學院數學系,山東德州 253023)

算子方程AX+X*B=C的解

趙琳琳

(德州學院數學系,山東德州 253023)

利用算子的廣義逆及相關投影,研究了一類算子方程的可解性,得到了方程可解的若干條件,并給出了解的一般表示.最后利用算子的矩陣表示,得到了此類算子方程可解的又一充要條件,進而豐富了這方面的研究.

Hilbert空間；算子方程；Moore-Penrose逆

1 引言

令H,K表示任意的Hilbert空間,B(H,K)表示從H到K的所有有界線性算子的集合,記B(H,H)=B(H).對給定的算子A∈B(H,K),B∈B(K,H),C∈B(K),本文研究算子方程

2 主要結果

顯然,(8)式有解等價于(12),(13)式有公共解.由引理2.1,得,

命題得證.

[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]田學剛,王少英.算子方程AX-XA*=B的正解與實正解[J].純粹數學與應用數學,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.

國家自然科學基金(10971070,11071079).

趙琳琳(1981-),博士,講師,研究方向：數值代數.

2010 MSC:47A62