就我們所知，有關算子方程AX-XB=C在有限維度上的理論，已經發展的很完善了。在這篇論文中，我們試著將近期有限維度上相關理論的發展，推廣到無窮維度上去。

As we know, the theory about the linear equation AX−XB=C has already been well developed in the finite-dimensional cases. In this paper, we will try to extend it to infinite-dimensional cases by using a similar technique developed recently in the finite-dimensional case.

Contents
1 Introduction 4
2 Equations of finite dimension - A survey 6
3 Equations of infinite dimension - A survey 11
4 Main results 14
References 17

[1] A. Jameson, ‘Solution of the equation AX −XB = C by inversion of an M ×M or N × N matrix’, SIAM J. Appl. Math. 16 (1968) 1020–1023.
[2] A. Schweinsberg, ‘The operator equation AX − XB = C with normal A and B’, Pacific J. Math. 102(1982) 447–453.
[3] E. Ma, ‘A finite series solution of the matrix equation AX−XB=C, SIAM J. Appl. Math. 14 (1966) 490–495.
[4] C. Davis and P. Rosenthal, ‘Solving linear operator equations’, Canad.J. Math. XXV I (1974) 1384–389.
[5] E. Heinz, ‘Beitr¨age zur St¨orungstheorie der Spektralzerlegung’, Math. Ann. 123 (1951) 415–438.
[6] E. Ma, ‘A finite series solution of the matrix equation AX−XB =C’,SIAM J. Appl. Math. 14 (1966) 490–495.
[7] G. Lumer and M. Rosenblum, ‘Linear operator equations’, Proc. Amer. Math. Soc. 10 (1959) 32–41.
[8] J. B. Conway, A course in functional analysis, second ed., Springer-Verlag New York, Inc. (1985).
[9] J. Sylvester, ‘Sur l’equation en matrices px = xq’, C. R. Acad. Sci. Paris 99 (1884) 67–71, 115–116.
[10] J.R.J. Jones, C. Lew, ‘Solutions to the Lyapunov matrix equation BX− XA = C’, IEEE Trans. Automat. Control AC-27 (2)(1982) 464–466.
[11] M. Rosenblum, ‘On the operator equation BX −XA = Q’, Duke Math. J. 23 (1956) 263–270.
[12] N. Lan, ‘On the Operator Equation AX−XB=C with unbounded Operators A, B and C’. Abstract and Applied Analysis June. 6, (2001) 317–328.
[13] P. Lancaster, M. Tismenetsky, The Theory of Matrices with Applications, second ed., Academic Press, New York, (1985).
[14] R. Bhatia, C.Davis and A. McIntosh, ‘Perturbation of spectral subspaces and solution of linear operator equations’, Linear Algebra Appl. 52/53
(1983) 45–67.
[15] R. Bhatia, C.Davis and P. Koosis, ‘An extremal problem in Fourier analysis with applications to operator theory’, J. Funct. Anal. 82 (1989) 138–150.
[16] R. Bahtia and P. Rosenthal, ‘How and why to solve the operator AX − XB = Y ’, Bull. Lond. Math. Soc., 29 (1997), 1–21.
[17] S. Barnett, C. Storey, Matrix Methods in Stability Theory, Nelson, London, (1970).
[18] W. E. Roth, ‘The equations AX − Y B = C and AX − XB = C in matrices’, Proc. Amer. Math. Soc.3 (1952) 392–396.
[19] W. Rudin, Real and complex analysis (3rd ed), McGraw-Hill, New York, (1991).