Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

Streszczenie

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