Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.00816

On the Shalit-Shamovich Spectral Radius

Maximilian Tornes
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that the spectral radius introduced by Shalit and Shamovich uniquely determines the underlying operator space structure: If E_1 and E_2 are operator space structures on C^d such that the corresponding spectral radii ρ_{E_1} and ρ_{E_2} agree on every d-tuple of operators, then the identity map is completely isometric. This answers a question of Scherer, Shalit and Shamovich. Moreover, we obtain an analogous result for the minimal spectral radii, and showcase an application to the completely bounded isomorphism problem for algebras of bounded non-commutative functions on operator space unit balls.

Explore related subjects

math.FAmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maximilian Tornes. 2026-09-30. On the Shalit-Shamovich Spectral Radius. https://arxiv.org/abs/2610.00816

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

KEEP EXPLORING

Related papers

arXiv2026

On hypercyclic and (common) upper frequently hypercyclic subspaces

Let $B$ be a unilateral weighted backward shift on $\ell_p$, $1 \leq p < \infty$, that admits a $\mathscr{U}$-frequently hypercyclic subspace. We prove that $B$ admits such a subspace free of frequently hypercyclic vectors. The proof technique we develop also allows us to prove that $B$ admits a hypercyclic subspace free of $\mathscr{U}$-frequently hypercyclic vectors, and to solve a question posed by Bès and Menet in 2015 on the existence of common $\mathscr{U}$-frequently hypercyclic subspaces.

math.FA↗
arXiv2026

Derivatives of Tensor Products and Applications to Spaces $C(K^n,X)$

In this paper, we develop an abstract theory of derivatives for Banach spaces based on objects that we call \emph{bidual assignments}. This framework encompasses both the Semadeni derivative and the recently introduced Semadeni--Pełczyński derivative. More generally, suitable ideals of subsets of dual spaces give rise to a broad family of derivatives within this setting. We establish direct-sum and tensor-product formulas for these derivatives, showing that they behave naturally with respect to direct sums and injective tensor products. We then obtain explicit descriptions of derivatives associated with compact trees and study the corresponding questions for finite products of compact lines. As the main application, we establish classification results for spaces of the form $C(K^n,X)$. In particular, for uncountable ordinals $α$ and $β$, an integer $n\geq1$, and Banach spaces $X$ satisfying suitable rigidity assumptions, we prove that \[ C([0,α]^n,X)\sim C([0,β]^n,X) \quad\text{if and only if}\quad C([0,α])\sim C([0,β]). \] This extends Kislyakov's classification of the spaces $C([0,α])$ and its vector-valued extension due to Galego to finite powers of ordinal intervals.

math.FA↗
arXiv2026

Nearly optimal Kolmogorov widths under holomorphic mappings

This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.

math.FA↗