Heinz inequality

In the sequel, a capital letter denotes a bounded linear operator on a Hilbert space . An operator is said to be positive (denoted by ) if for all . In 1951, E. Heinz [a3] proved a series of very useful norm inequalities; one of the most essential inequalities in operator theory being:

(a1)

where and are positive operators and .

It is shown in [a1] and [a2] that the Heinz inequality (a1) is equivalent to each of the inequalities (a2), (a3) and (a4). Other norm inequalities equivalent to (a1) have also been obtained in [a1] and [a2]. For any operators , and ,

(a2)

For a self-adjoint and invertible operator ,

(a3)

For and self-adjoint ,

(a4)

The inequality (a2) has been obtained in [a4] to give an alternative ingenious proof of (a1). The original proof of the Heinz inequality (a1), based on deep calculations in complex analysis, is shown in [a3]; a simplified and elementary proof of (a1) is given in [a2].

See also Heinz–Kato inequality; Heinz–Kato–Furuta inequality.

References

[a1]	J.I. Fujii, M. Fujii, T. Furuta, R. Nakamoto, "Norm inequalities related to McIntosh type inequality" Nihonkai Math. J. , 3 (1992) pp. 67–72
[a2]	J.I. Fujii, M. Fujii, T. Furuta, R. Nakamoto, "Norm inequalities equivalent to Heinz inequality" Proc. Amer. Math. Soc. , 118 (1993) pp. 827–830
[a3]	E. Heinz, "Beiträge zur Störungstheorie der Spektralzerlegung" Math. Ann. , 123 (1951) pp. 415–438
[a4]	A. McIntosh, "Heinz inequalities and perturbation of spectral families" Macquarie Math. Reports (1979) pp. unpublished