Young’s (in)equality for compact operators

Tom 233 / 2016

Gabriel Larotonda Studia Mathematica 233 (2016), 169-181 MSC: Primary 15A45, 47A30; Secondary 15A42, 47A63. DOI: 10.4064/sm8427-5-2016 Opublikowany online: 19 May 2016


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 $|a|^p=|b|^q$.


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

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