A simplified proof of Heinz inequality and scrutiny of its equality

Author:
Takayuki Furuta

Journal:
Proc. Amer. Math. Soc. **97** (1986), 751-753

MSC:
Primary 47A30

DOI:
https://doi.org/10.1090/S0002-9939-1986-0846001-3

MathSciNet review:
846001

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Abstract | References | Similar Articles | Additional Information

Abstract: An operator means a bounded linear operator on a Hilbert space $H$. We give a simplified proof of the following inequality: \[ ({{\text {I}}_1})\quad |(Tx,y){|^2} \leq (|T{|^{2\alpha }}x,x)(|{T^*}{|^{2(1 - \alpha )}}y,y)\] for any operator $T$ and for any $x, y \in H$ and for any real number $\alpha$ with $0 \leq \alpha \leq 1$. *In case* $0 < \alpha < 1$, *the equality in* $({{\text {I}}_1})$ *holds iff* $|T{|^{2\alpha }}x$ *and* ${T^*}y$ *are linearly dependent iff* $Tx$ *and* $|{T^*}{|^{2(1 - \alpha )}}y$ *are linearly dependent*. $({{\text {I}}_1})$ is equivalent to \[ ({{\text {I}}_2})\quad |(Tx,y)| \leq ||\;|T{|^\alpha }x||\;||\;|{T^*}{|^{1 - \alpha }}y||,\], so one might believe that the equality in $({{\text {I}}_1})$ or $({{\text {I}}_2})$ would hold iff $|T{|^{2\alpha }}x$ and $|{T^*}{|^{2(1 - \alpha )}}y$ are linearly dependent or iff $|T{|^\alpha }x$ and $|{T^*}{|^{1 - \alpha }}y$ are linearly dependent, but we can give counterexamples to these mistakes. By this fact, the form of $({{\text {I}}_1})$ is more convenient than $({{\text {I}}_2})$ in order to remind us of the case when the equality in $({{\text {I}}_1})$ or $({{\text {I}}_2})$ holds.

- J. Dixmier,
*Sur une inégalité de E. Heinz*, Math. Ann.**126**(1953), 75–78 (French). MR**56200**, DOI https://doi.org/10.1007/BF01343151 - Erhard Heinz,
*On an inequality for linear operators in a Hilbert space*, Report of an international conference on operator theory and group representations, Arden House, Harriman, N. Y., 1955, Publ. 387, National Academy of Sciences-National Research Council, Washington, D. C., 1955, pp. 27–29. MR**0079139** - Tosio Kato,
*Notes on some inequalities for linear operators*, Math. Ann.**125**(1952), 208–212. MR**53390**, DOI https://doi.org/10.1007/BF01343117

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Keywords:
Heinz inequality,
polar decomposition

Article copyright:
© Copyright 1986
American Mathematical Society