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Young’s inequality in compact operators. (English) Zbl 1031.47013

Gohberg, Israel (ed.) et al., Linear operators and matrices. The Peter Lancaster anniversary volume. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 130, 171-184 (2002).
In this paper, the authors prove Young’s inequality for compact operators as follows: If \(a\) and \(b\) are compact operators acting on a complex separable Hilbert space, then there is a partial isometry \(u\) such that the initial space of \(u\) is \((\ker |ab^{*}|)^{\perp}\) and \[ u|ab^{*}|u^{*}\leq \frac{1}{p}|a|^{p}+\frac{1}{q}|b|^{q} \] for any \(p,q\in (0,\infty)\) that satisfy \(\frac{1}{p}+\frac{1}{q}=1\). Furthermore, if \(|ab^{*}|\) is injective, then the operator \(u\) in the inequality above can be taken to be a unitary. This is an extension of the matrix mean inequality [SIAM J. Matrix Anal. Appl. 11, 272-277 (1990; Zbl 0704.47014)] and Ando’s result [Oper. Theory, Adv. Appl. 75, 33-38 (1994; Zbl 0830.47010)] which considered the above inequality for complex matrices.
For the entire collection see [Zbl 1005.00035].

MSC:

47A63 Linear operator inequalities
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