Tensor factorization and explicit spectral bounds for product-box concentration operators

Ahmadreza Azimifard

functional analysis

Tensor factorization and explicit spectral bounds for product-box concentration operators

arXiv:2607.26361

summary

The paper derives explicit uniform upper bounds on the number of eigenvalues of spatio‑spectral concentration operators that lie away from 0 and 1 when the underlying sets are finite unions of axis‑parallel boxes, using a tensor‑factorization approach and providing trace asymptotics for these operators.

Abstract

Let $S = P_{c A_{0}} Q_{B_{0}} P_{c A_{0}}$ be the spatio-spectral concentration operator of bounded sets $c A_{0}, B_{0} \subset R^{d}$ , and let $I^_{ε} = # {n : ε < I^»_{n} (S) < 1 - ε}$ be its plunge count. For $A_{0}$ and $B_{0}$ finite disjoint unions of bounded axis-parallel open boxes, we prove an explicit uniform upper bound on $I^_{ε}$ , valid for every $d \geq 1$ , $c > 0$ , and $0 < ε < 1/2$ , with all constants written in terms of the side lengths. On the range $I^\pm \geq 4$ , $c \geq 2$ , and $I^\pm^{- c} < ε < 1/2$ , it gives $I^_{ε} \leq C c^{d - 1} lo g (1/ ε) lo g (I^\pm c / lo g (1/ ε))$ . Kulikov and Dam Larsen previously proved this order on that range for a broader class; the present contribution is an independent proof and an explicit all-parameter estimate for product boxes. The proof uses a telescoping tensorization of $P_{(c A_{0})^{c}} Q_{B_{0}} P_{c A_{0}}$ into $d$ elementary tensor operators, with one one-dimensional off-diagonal factor and $d - 1$ localization factors. Schatten quasi-norms then multiply across tensor factors, and the single logarithm arises only from the normal direction. For the model cube pair, we also prove that, when $ε < 4^{- d}$ , $I^_{ε} \geq M_{a}^{d} = I^c ◯ ((lo g c)^{d})$ . Using an exact trace identity, an explicit cubic minorant, and the sine-kernel determinant asymptotics of Basor and Widom, we further obtain $Tr ((S - S^{2})^{m}) = I^_{m}^{2} I ¨ ^{- 2} lo g c + O_{m} (1)$ for each fixed $m$ , where $I^_{m}^{2} = B (m, m)$ , together with a two-sided fixed-depth window estimate of order $lo g c$ . The lower bound is not matching, and the fixed-order statements are not uniform in $m$ .

36 pages. Explicit spectral bounds for finite unions of axis-parallel product boxes; includes tensor-factorization arguments, cube-model lower bounds, and fixed-order trace asymptotics

Topics & keywords

#spectral concentration #product boxes #tensor factorization #eigenvalue bounds #trace asymptotics #Schatten normsconcentration operatorplunge counttensorizationaxis‑parallel boxesSchatten quasi‑normsine‑kernel determinant