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Further Extension of the Heinz-Kato-Furuta Inequality
Proceedings of the American Mathematical Society
Vol. 127, No. 10 (Oct., 1999), pp. 2899-2904
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/119946
Page Count: 6
You can always find the topics here!Topics: Monotonic functions, Mathematical inequalities, Hilbert spaces, Mathematical theorems, Continuous functions
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Let T be a bounded operator on a Hilbert space H, and A, B positive definite operators. Kato has shown that if |Tx|≤ |Ax| and |T*y|≤ |By| for all x, y ∈ H, then |(Tx,y)|≤ |f(A)x||g(B)y|, where f(t), g(t) are operator monotone functions defined on [0, ∞ ) such that f(t)g(t) = t. Furuta has shown that |(T|T|α +β -1x,y)|≤ |Aαx||Bβy|, where 0 ≤ α,β ≤ 1, 1 ≤ α +β . Let f(t), g(t) be any continuous operator monotone functions, and set h(t) = f(t)g(t)/t for t > 0. We will show that Th(|T|) is well defined and |(Th(|T|)x,y)|≤ |f(A)x||g(B)y|. Moreover, we will extend this result for unbounded closed operators densely defined on H.
Proceedings of the American Mathematical Society © 1999 American Mathematical Society