Abstract
An estimate for the norm of the solution to the equation AX−XB=S obtained by R. Bhatia, C. Davis, and A. McIntosh for normal operators A and B is shown to be valid for a larger class. Some other inequalities in the same spirit are obtained, including a “sinθ theorem” for singular vectors. Some inequalities concerning the continuity of the map A→|itA| obtained recently by Kittaneh and Kosaki are extended using these ideas.
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