It has been proved by Koehler and Rosenthal [Studia Math. 36 (1970), 213–216] that an linear isometry $$ preserves some semi-inner-product. Recently, similar investigations have been carried out by Niemiec and Wójcik for continuous representations of amenable semigroups into $$ (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 $$ of norm one attains its norm then there is a semi-inner-product $$ that the operator $$ preserves this semi-inner-product on the norm attaining set. More precisely, we show that the equality $$ holds for all $$.