Abstract
It is shown that ifA, B andX are operators on a Hilbert space such thatA andB are positive andX belongs to a norm ideal associated with some unitarily invariant norm |‖·|‖, then for 0 ≤r ≤ 1 we have |‖A r XB r|‖ ≤ |‖X|‖1-r|‖AXB|‖r. This is an extension of the classical Heinz-Kato inequality which was originally proved for the usual operator norm. Other related inequalities are also discussed.
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References
Araki, H., On an inequality of Lieb and Thirring,Lett. Math. Phys. 19, 167–170 (1990).
Bhatia, R., Perturbation inequalities for the absolute value map in norm ideals of operators,J. Operator Theory 19, 129–136 (1988).
Bhatia, R. and Kittaneh, F., Norm inequalities for partitioned operators and an application,Math. Ann. 287, 719–726 (1990).
Cordes, H. O.,Spectral Theory of Linear Differential Operators and Comparison Algebras, London Math. Soc. Lecture Note Ser. 76, Cambridge Univ. Press, 1987.
Fujii, J. I. and Fujii, M., A norm inequality for operator monotone functions,Math. Japon. 35, 249–252 (1990)
Furuta, T., Norm inequalities equivalent to Löwner-Heinz theorem,Rev. Math. Phys. 1, 135–137 (1989).
Furuta, T. and Hakeda, J., Applications of norm inequalities equivalent to Löwner-Heinz theorem,Nihonkai Math. J. 1, 11–17 (1990).
Gohberg, I. C. and Kerin, M. G.,Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, Amer. Math. Soc. Providence, R.I., 1969.
Halmos, P. R.,A Hilbert Space Problem Book, Springer, New York, 1982.
Heinz, E., Beiträge zur Störungstheorie der Spektralzerlegung,Math. Ann. 123, 415–438 (1951).
Horn, R. A. and Mathias, R., Cauchy-Schwarz inequalities associated with positive semidefinite matrices,Linear Algebra Appl. 142, 63–82 (1990).
Kato, T., A generalization of the Heinz inequality,Proc. Japan. Acad. Ser. A. Math. Sci. 6, 305–308 (1961).
Kittaneh, F. and Kosaki, H., Inequalities for the Schattenp-norm V,Publ. RIMS, Kyoto Univ. 23, 433–443 (1987).
McIntosh, A., Heinz inequalities and perturbation of spectral families, Macquarie Mathematics Reports, Macquarie University, 1979.
Schatten, R.,Norm Ideals of Completely Continuous Operators, Springer, Berlin, 1960.
Simon, B.,Trace Ideals and their Applications, Cambridge University Press, Cambridge, 1979.