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 $n\geq 2$, $$|S_α^{l_1}\odot\cdots \odot S_α^{l_k}\odot S_α^{*l_{k+1}}\odot\cdots \odot S_α^{*l_{n}}| =\mathop{\prod}{i=1}^n\left | S_α^{l{i}}\right|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ 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}.

💡 Research Summary

The paper investigates the operator norm of symmetric and antisymmetric tensor products of weighted shift operators on a Hilbert space. Let (S_{\alpha}) denote a unilateral weighted shift with weight sequence (\alpha={\alpha_i}{i\ge0}), and let (S{\alpha}^{*}) be its adjoint. The authors introduce the notion of regularity for the weight sequence: (\alpha) is regular if (\lim_{i\to\infty}|\alpha_i|=\lambda) exists and dominates all earlier weights, i.e. (|\alpha_i|\le\lambda) for every (i). This condition is satisfied by virtually all classical weighted shifts (Hardy shift, Bergman shift, Dirichlet shift, etc.).

The main result (Theorem 1.3) states that for any integer (n\ge2) and any non‑negative integer exponents ((l_1,\dots,l_n)), \