Young’s (in)equality for compact operators

Gabriel Larotonda ¹

¹ Instituto de Ciencias Universidad Nacional de General Sarmiento J. M. Gutierrez 1150 (B1613GSX) Los Polvorines Buenos Aires, Argentina

Studia Mathematica, Tome 233 (2016) no. 2, pp. 169-181

Résumé

If $a,b$ are $n\times n$ matrices, T. Ando proved that Young’s inequality is valid for their singular values: if $p \gt 1$ and $1/p+1/q=1$, then $$ \lambda_k(|ab^*|)\le \lambda_k\biggl( \frac1p |a|^p+\frac 1q |b|^q \biggr) \quad\ \text{for all } k. $$ Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if $a,b$ are compact operators, then equality holds in Young’s inequality if and only if $|a|^p=|b|^q$.

Détail

DOI : 10.4064/sm8427-5-2016

Keywords: times matrices ando proved young inequality valid their singular values lambda * lambda biggl frac frac biggr quad text later result extended singular values pair compact operators acting hilbert space erlijman farenick zeng paper prove compact operators equality holds young inequality only