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

TL;DR

The paper investigates norms of symmetric and antisymmetric tensor products of weighted shift operators $S_\alpha$ on a Hilbert space. It introduces a regularity condition on the weights and proves an equivalence: for all integers $n\ge 2$ and all multi-indices $(l_1,\dots,l_n)\in\mathbb{N}^n$, the norm of the tensor product equals the product of the individual norms if and only if the weights are regular; in that regime the common norm is $\lambda^{\sum_i l_i}$ with $\lambda=\lim|\alpha_i|$, and for many standard shifts this reduces to unity. The results partially solve Problems 6, 7, 1, and 2 posed by Garcia and colleagues, and demonstrate that classical shifts (Hardy, Bergman, Dirichlet, etc.) satisfy the regularity condition, enabling exact norm computations for symmetric and antisymmetric tensor products. Additionally, the paper establishes universal lower bounds for symmetric tensor products, providing foundational inequalities for both vector and operator cases and reinforcing the sharpness of the derived norm equalities.

Abstract

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

Theorems & Definitions (33)

• Definition 1.1
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof