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A ≥ B ≥ 0 Assures (Br Ap Br)1/q ≥ B^{(p+2r)/q for r ≥ 0, p ≥ 0, q ≥ 1 with (1 + 2r)q ≥ p + 2r

Takayuki Furuta
Proceedings of the American Mathematical Society
Vol. 101, No. 1 (Sep., 1987), pp. 85-88
DOI: 10.2307/2046555
Stable URL: http://www.jstor.org/stable/2046555
Page Count: 4
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A ≥ B ≥ 0 Assures (Br Ap Br)1/q ≥ B^{(p+2r)/q for r ≥ 0, p ≥ 0, q ≥ 1 with (1 + 2r)q ≥ p + 2r
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Abstract

An operator means a bounded linear operator on a Hilbert space. This paper proves the assertion made in its title. Theorem 1 yields the famous result that A ≥ B ≥ 0 assures Aα ≥ Bα for each α ∈ [ 0, 1 ] when we put r = 0 in Theorem 1. Also Corollary 1 implies that A ≥ B ≥ 0 assures (BApB)1/p ≥ B(p + 2)/p for each p ≥ 1 and this inequality for p = 2 is just an affirmative answer to a conjecture posed by Chan and Kwong. We cite three counterexamples related to Theorem 1 and Corollary 1.

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