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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.

MSC:

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

Citations:

Zbl 1254.47015
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References:

[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
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