Heinz inequality

In the sequel, a capital letter denotes a bounded linear operator on a Hilbert space $H$. An operator $T$ is said to be positive (denoted by $T\geq0$) if $(Tx,x)\geq0$ for all $x\in H$. In 1951, E. Heinz [a3] proved a series of very useful norm inequalities; one of the most essential inequalities in operator theory being:

$$\|S_1Q+QS_2\|\geq\|S_1^\alpha QS_2^{1-\alpha}+S_1^{1-\alpha}QS_2^\alpha\|,\label{a1}\tag{a1}$$

where $S_1$ and $S_2$ are positive operators and $1\geq\alpha\geq0$.

It is shown in [a1] and [a2] that the Heinz inequality \eqref{a1} is equivalent to each of the inequalities \eqref{a2}, \eqref{a3} and \eqref{a4}. Other norm inequalities equivalent to \eqref{a1} have also been obtained in [a1] and [a2]. For any operators $P$, $Q$ and $R$,

$$\|P^*PQ+QRR^*\|\geq2\|PQR\|.\label{a2}\tag{a2}$$

For a self-adjoint and invertible operator $S$,

$$\|STS^{-1}+S^{-1}TS\|\geq2\|T\|.\label{a3}\tag{a3}$$

For $A\geq0$ and self-adjoint $Q$,

$$\|\operatorname{Re}A^2Q\|\geq\|AQA\|.\label{a4}\tag{a4}$$

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 unpublished