# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A simplified proof of Heinz inequality and scrutiny of its equalityHTML articles powered by AMS MathViewer

by Takayuki Furuta
Proc. Amer. Math. Soc. 97 (1986), 751-753 Request permission

## 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.
References
• J. Dixmier, Sur une inégalité de E. Heinz, Math. Ann. 126 (1953), 75–78 (French). MR 56200, DOI 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, National Academy of Sciences-National Research Council, Washington, D.C., 1955, pp. 27–29. Publ. 387. MR 0079139
• Tosio Kato, Notes on some inequalities for linear operators, Math. Ann. 125 (1952), 208–212. MR 53390, DOI 10.1007/BF01343117
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