Solvability of \(AXB - CXD = E\) in the operators algebra \(B(H)\). (English) Zbl 1300.47023

This short note is a continuation of the paper published by the author and S. Mecheri [Lobachevskii J. Math. 30, No. 3, 224–228 (2009; Zbl 1254.47015)]. It is proved that the equation \(AXB-CXD=E\), where \(A, B, C, D\in B(H)\) are normal operators on a separable (infinite-dimensional) complex Hilbert space \(H\) such that \(A\) and \(D\) commute and the pairs \((A,D)\) and \((B,C)\) possess the Fuglede-Putnam property, has a solution in \(B(H)\) if and only if, for every \(\lambda \in {\mathbb C}\), the block operator matrix which has \(A-\lambda I\) and \(D-\lambda I\) on the diagonal is equivalent to the block operator matrix which has \(A-\lambda I\) and \(D-\lambda I\) on the diagonal and \(E\) in the upper right corner.


47A62 Equations involving linear operators, with operator unknowns
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B47 Commutators, derivations, elementary operators, etc.


Zbl 1254.47015
Full Text: DOI


[1] J. H. Anderson and C. Foias, Properties which normal operator share with normal derivation and related operators, Pacific J. Math. 61, 313 (1976). · Zbl 0324.47018
[2] A. Bachir and A. Sagres, A generalized Fuglede-Putnam theorem and orthogonality, Aust. J. Math. Anal. Appl. 1(1), Art. 12, 5 (2004) (electronic). · Zbl 1084.47017
[3] R. Bhatia, Matrix Analysis (Springer-Verlag, New York, 1997), Graduate Texts in Mathematics.
[4] J. B. Conway,A Course in OperatorTheory, (American Mathematical Society, Providence, R.I, 1999), Graduate Studies in Mathematics, Vol. 21.
[5] B. P. Duggal, Putnam-Fuglede theorem and the range-kernel orthogonality of derivations, I. JMMS 27(9), 573 (2001). · Zbl 1010.47013
[6] H. Flanders and H. K. Wimmer, On the matrix equation AX - XB = C and AX - YB = C, SIAM. J. Appl. Math. 32, 707 (1977). · Zbl 0385.15008
[7] P. R. Halmos, A Hilbert Space Problem Book, D. Van Nostrand Company (Inc. Princeton, New Jerse, 1967).
[8] P. J. Maher, Commutator approximants, Proc. Amer. Math. Soc. 115, 995 (1992). · Zbl 0773.47020
[9] S. Mecheri, Another version of Maher’s inequality, Zeitschrift fr Analysis und ihre Andwendungen 23(2), 303 (2004). · Zbl 1085.47045
[10] S. Mecheri and A. Mansour, On the operator equation AXB - XD = E, Lobachevskii journal of mathematics 30(NTA3), 224 (2009). · Zbl 1254.47015
[11] M. Resenblum, On the operator equation AX - XB = Q, Duke. Math. J. 23, 263 (1956). · Zbl 0073.33003
[12] M. Resenblum, On the operator equation AX - XB = Q with self-adjoint A and B, Proc. Amer. Math. Soc. 20, 115 (1969).
[13] W. E. Roth, The equations AX - YB = Qand AX - XB = Qinmatrices, Proc. Amer. Math. Soc. 3, 302 (1952).
[14] H. Wielandt, ber die Unbeschrnktheit der Operatoren der Quanten-mechanik, (German) Math. Ann. 121, 21 (1949). · Zbl 0035.19903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.