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Proceedings of the American Mathematical Society

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

 

 

Two mixed Hadamard type generalizations of Heinz inequality


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 103 (1988), 91-96
MSC: Primary 47A30; Secondary 47A05
DOI: https://doi.org/10.1090/S0002-9939-1988-0938650-0
MathSciNet review: 938650
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Abstract: We give two types of mixed Hadamard inequalities containing the terms $ T,\left\vert T \right\vert$, and $ \left\vert {{T^ * }} \right\vert$, where $ T$ is a bounded linear operator on a complex Hilbert space. As an immediate consequence of these results, we can easily show some extensions of the Hadamard inequality and also the Heinze inequality:

$\displaystyle \left( * \right)\quad {\left\vert {\left( {Tx,y} \right)} \right\... ...{\left\vert {{T^ * }} \right\vert}^{2\left( {1 - \alpha } \right)}}y,y} \right)$

for any $ T$, any $ x,y$ in $ H$, and any real number $ \alpha$ with $ 0 \leq \alpha \leq 1$. And the following conditions are equivalent in case $ 0 < \alpha < 1$:

(1) the equality in (*) holds;

(2) $ {\left\vert T \right\vert^{2\alpha }}x$ and $ {T^ * }y$ are linearly dependent;

(3) $ Tx$ and $ {\left\vert {{T^ * }} \right\vert^{2\left( {1 - \alpha } \right)}}y$ are linearly dependent.

Results in this paper would remain valid for unbounded operators under slight modifications.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938650-0
Keywords: Heinz inequality, Hadamard inequality, mixed Schwarz inequality, polar decomposition
Article copyright: © Copyright 1988 American Mathematical Society