Semi-inner products and operators which attain their norm

Abstract

It has been proved by Koehler and Rosenthal [Studia Math. 36 (1970), 213–216] that an linear isometry $U\in\mathcal{L}(X)$ preserves some semi-inner-product. Recently, similar investigations have been carried out by Niemiec and Wójcik for continuous representations of amenable semigroups into $\mathcal{L}(X)$ (cf. [Studia Math. 252 (2020), 27–48]). In this paper we generalize the result of Koehler and Rosenthal. Namely, we prove that if an operator $T\in \mathcal{L}(X)$ of norm one attains its norm then there is a semi-inner-product $[\cdot|\diamond]: X\times X\to \mathbf{F}$ that the operator $T$ preserves this semi-inner-product on the norm attaining set. More precisely, we show that the equality $[T(\cdot)|\mathit{Tx}]=[\cdot|x]$ holds for all $x\in M_{T}:=\{y\in S_{X}: \|\mathit{Ty}\|=1\}$.