Further Results on the Isometric Embeddability of $S_q^m$ into $S_p^n$

Ying Xu; Wenwen Zhang; Qi Liu

Home/Archive/arXiv

arXiv · 2609.37504

Further Results on the Isometric Embeddability of $S_q^m$ into $S_p^n$

Ying Xu·Wenwen Zhang·Qi Liu

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study Schatten embeddings through circular means of trace powers. In finite dimensions, a decomposition by singular-value vanishing orders shows that every term of order at most two has a nonnegative coefficient. This yields the classification $\ell_q^2(\C)\hookrightarrow\Sp_p^n$ if and only if $q=p$ or $q=2$, for $n\ge2$ and $p<\infty$. In particular, it gives negative answers to parts (ii), (iii), and (iv) of Problem 3.1 in Chattopadhyay--Pradhan--Skripka, arXiv:2603.07359v2. A regularized curvature estimate also excludes $\ell_q^2$ from infinite-dimensional $\Sp_p$ for $0<p\le2<q\le\infty$ and gives finite-rank distortion bounds. The other parts of that problem are not claimed to be solved.Our results extend previous studies on Schatten embeddings, particularly those in \cite{CHPPR, CHPR, CPS}, by developing a complex-curvature framework based on complex convexity theories \cite{BR,DGT}, which resolves several remaining quasi-Banach cases and yields new geometric obstructions for isometric embeddings.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ying Xu, Wenwen Zhang, Qi Liu. 2026-09-26. Further Results on the Isometric Embeddability of $S_q^m$ into $S_p^n$. https://arxiv.org/abs/2609.37504

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric

We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form $[\det(I-A_i^*A_j)^{-α}]_{ij}$, where $A_i$ and $A_j$ are strict contractions and $α\in\mathbb{R}$). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which $\det(I-A^*B)^{-α}$ is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.

math.FA↗

arXiv2026

Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor

Leveraging tools from convex analysis and incorporating additional singular value information of matrices, we completely resolve the problem of establishing perturbation bounds for the Frobenius norm of subunitary and positive polar factors. We derive corresponding sharp upper and lower bounds. As corollaries, we refine the results of Li and Sun [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1183--1193] and strengthen the classical Araki-Yamagami inequality [Comm. Math. Phys., 81 (1981), no. 1, pp. 89--96]. The versatility of our method also allows us to strengthen Lee's conjecture, providing a sharper version along with a matching sharp lower bound. Furthermore, we generalize the classical matrix arithmetic-geometric mean inequality and Cauchy-Schwarz inequality into tighter and more robust forms. Finally, we establish a sharp lower bound for a result by Kittaneh [Comm. Math. Phys., 104 (1986), no. 2, pp. 307--310].

math.FA↗

arXiv2026

On the Dilation Theory and Canonical Decomposition of $\mathbfΘ_n$-Contractions

This paper investigates the domain $\mathbfΘ_n$ from an operator-theoretic perspective. We establish several characterizations of $\mathbfΘ_n$-contractions, $\mathbfΘ_n$-unitaries, and $\mathbfΘ_n$-isometries, and explore their connections with $Γ_n$- and tetrablock operator tuples, as well as with the corresponding operator classes associated with $\mathbfΘ_{n+1}$. We prove that every $\mathbfΘ_n$-contraction admits a canonical decomposition as the direct sum of a $\mathbfΘ_n$-unitary and a completely non-unitary $\mathbfΘ_n$-contraction. We further develop the conditional dilation theory for $\mathbfΘ_n$-contractions and establish necessary conditions for the existence of $\mathbfΘ_n$-isometric dilations. We show that the conditions in Theorem~\ref{Conditional Dilation} are not, in general, sufficient by constructing a $\mathbfΘ_3$-contraction that admits a $\mathbfΘ_3$-isometric dilation although condition~(2) of Theorem~\ref{Conditional Dilation} fails. We further study pure $\mathbfΘ_n$-isometric dilations of $\mathbfΘ_n$-contractions and establish a Hardy space characterization of such dilations. We show that the resulting $\mathbfΘ_n$-isometric dilation is minimal. We also study isometric dilations of doubly commuting $\mathbfΘ_n$-contractions under suitable Hardy space compatibility conditions. Finally, we identify a special class of $\mathbfΘ_2$-contractions that admits a $\mathbfΘ_2$-isometric dilation.

math.FA↗