Young's (in)equality for compact operators (Q2809361)

Young's inequality asserts that if \(p>1\) and \(1/p+1/q=1\), then for any \(\alpha, \beta\in \mathbb R^+\), NEWLINE\[NEWLINE \alpha\beta\leq \frac 1p \alpha^p + \frac 1q \beta ^q NEWLINE\]NEWLINE with equality if and only if \(\alpha^p=\beta^q\). \textit{T. Ando} [in: Operator theory in function spaces and Banach lattices. Essays dedicated to A. C. Zaanen on the occasion of his 80th birthday. Symposium, Univ. of Leiden, NL, 1993. Basel: Birkhäuser. 33--38 (1994; Zbl 0830.47010)] proved that Young's inequality is valid for the singular values of two \(n\times n\) matrices, which was extended to the singular values of a pair of compact operators acting on a Hilbert space by \textit{J. Erlijman} et al. [in: Linear operators and matrices. The Peter Lancaster anniversary volume. Basel: Birkhäuser. 171--184 (2002; Zbl 1031.47013)]: if \(a, b\) are the compact operators, then for all \(k\in \mathbb N_0\), NEWLINE\[NEWLINE \lambda_{k}(|ab^{*}|)\leq \lambda_{k}\left(\frac{1}{p}|a|^{p}+\frac{1}{q}|b|^{q}\right), NEWLINE\]NEWLINE where \(|y| = (y^*y)^{1/2}\) denotes the positive operator in the polar decomposition of \(y=\nu|y|\), and \(\nu\) is a partial isometry. Here, \(\lambda_k(x)\) denotes the \(k\)-th eigenvalue of the positive compact operator, arranged in non-increasing order. This paper proves that, if \(a,b\) are compact operators, then the equality NEWLINE\[NEWLINE \lambda_{k}(|ab^{*}|)=\lambda_{k}\left(\frac{1}{p}|a|^{p}+\frac{1}{q}|b|^{q}\right) NEWLINE\]NEWLINE holds for all \(k\in \mathbb N_0\) if and only if \(|a|^{p}=|b|^{q}.\)