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

In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\ge 2$, $$\mathrm{\parallel }{S}_{\alpha }^{l_1}\odot \cdots \odot S_{\alpha }^{l_k}\odot S_{\alpha }^{\ast l_{k+1}}\odot \cdots \odot S_{\alpha }^{\ast l_n}\mathrm{\parallel }={}_{i=1}^n\parallel S_{\alpha }^{l_i}\parallel ,\text{ for any }(l_1,l_2\cdots l_n)\in {\mathbb{N}}^n$$ if and only if the weight satisfies the regularity condition, we partially solve \cite[Problem 6 and Problem 7]{GA}. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, Dirichlet shift, etc, satisfy the regularity condition. Moreover, at the end of the paper, we solve \cite[Problem 1 and Problem 2]{GA}.