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

TL;DR

This paper investigates the norms of symmetric and antisymmetric tensor products of weighted shift operators, establishing conditions under which these norms factorize, and applies these results to various classical weighted shift operators.

Contribution

It provides a characterization of when the tensor product norms factorize for weighted shift operators, solving specific open problems in the field.

Findings

Norms factorize under regularity condition for weights

Most classical weighted shifts satisfy the regularity condition

Partial solutions to two open problems in the literature

Abstract

In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$, $$\|S_{\alpha}^{l_1}\odot\cdots \odot S_{\alpha}^{l_k}\odot S_{\alpha}^{*l_{k+1}}\odot\cdots \odot S_{\alpha}^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_{\alpha}^{{l_{i}}}\right\|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ 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}.