Symmetric and Antisymmetric Tensor Products for the Function-Theoretic Operator Theorist

TL;DR

The paper investigates norms and spectra of symmetric and antisymmetric tensor powers of bounded Hilbert-space operators. It builds a foundational framework by defining tensor-power spaces $\mathcal{H}^{\odot n}$ and $\mathcal{H}^{\wedge n}$ via the projection operators ${\mathrm S}_n$ and ${\mathrm A}_n$, and then analyzes symmetric/antisymmetric products $A_1 \odot \cdots \odot A_n$ and $A_1 \wedge \cdots \wedge A_n$ in general, establishing key properties and adjoint relations. It provides concrete results for important classes, including the norm and spectral radius relations $\|A^{\odot n}\|=\|A\|^n$ and $\rho(A^{\odot n})=\rho(A)^n$, exact spectra in special cases such as diagonal operators and the shift $S$ with its adjoint, and sharp norm bounds for diagonal factors. The work also demonstrates rich spectral phenomena, such as spectra that can have positive measure, and culminates with numerous open questions that chart a course for further exploration in function-theoretic operator theory and related areas.

Abstract

We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

Figures (1)

• Figure 1: Colors denote the step where the $a_{k,\ell}$ are fixed: Step (1) is in violet; (2) is in red; (3) is in green; and (4) is in blue. The symmetry of symmetric tensors permits us to focus on $\ell \geqslant k \geqslant 0$. The violet and red values are zero.

Theorems & Definitions (63)

• Definition 2.2: Tensor powers of Hilbert spaces
• Proposition 2.4
• proof
• Definition 2.5: Symmetric and antisymmetric tensor powers of Hilbert spaces
• Example 2.6
• Definition 2.9: Symmetrization and antisymmetrization operators
• Proposition 2.11
• proof
• Proposition 2.12
• Proposition 2.13