The norms for symmetric and antisymmetric tensor products of the weighted shift operators

Abstract In the present article, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$ , $$ \begin{align*}\|S_{\alpha}^{l_1}\odot\cdots \odot S_{\alpha}^{l_k}\odot S_{\alpha}^{*l_{k+1}}\odot\cdots \odot S_{\alpha}^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_{\alpha}^{{l_{i}}}\right\|, \text{ for any} \ (l_1,l_2\ldots l_n)\in\mathbb N^n\end{align*} $$ if and only if the weight satisfies the regularity condition, we partially solve [see Problems 6 and 7 in Garcia et al. (2025, Canad. J. Math. , 77, 324–346)]. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, etc., satisfy the regularity condition. Moreover, at the end of the article, we solve [see Problems 1 and 2 in Garcia et al. (2025)].