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About 10,000 papers for “Operator space”

Operator space tensor norms

Alejandro Chávez-Domínguez, Verónica Dimant, Daniel Galicer · arXiv (Cornell University) · 2023

The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. The aim of this manuscript is to kick off the development of a systematic theory of tensor products and tensor norms for operator spaces and its interplay with their associated mapping ideals. Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definitions in the operator space framework. The theory is not a mere translation of what is known in the classical setting and new insights, techniques, ideas or hypotheses are required in many cases. As a consequence, notable differences in the theory appear when compared to the classical one.

Cited by 0Open access

Operator Space Complexification Transfigured

David P. Blecher, Mehrdad Kalantar · International Mathematics Research Notices · 2024

Abstract Given a finite group $G$, a central subgroup $H$ of $G$, and an operator space $X$ equipped with an action of $H$ by complete isometries, we construct an operator space $X_{G}$ equipped with an action of $G$ that is unique under a “reasonable” condition. This generalizes the operator space complexification $X_{c}$ of $X$. As a linear space $X_{G}$ is the space obtained from inducing the representation of $H$ to $G$ (in the sense of Frobenius). Indeed a main achievement of our paper is the induced representation construction in the category of operator spaces. This has been hitherto elusive even for Banach spaces since it is not clear how to norm the induced space. We show that for a large class of group actions the induced space has a unique operator space norm.

Cited by 4

THE OPERATOR SHIFT SPACE

Timur Oikhberg · Proceedings of the Edinburgh Mathematical Society · 2008

Abstract We construct and examine an operator space $X$, isometric to $\ell_2$, such that every completely bounded map from its subspace $Y$ into $X$ is a compact perturbation of a linear combination of multiples of a shift of given multiplicity and their adjoints. Moreover, every completely bounded map on $X$ is a Hilbert–Schmidt perturbation of such a linear combination.

Cited by 2Open access

An Operator Space Bestiary

Vern I. Paulsen · Cambridge University Press eBooks · 2003

In the last chapter we obtained an abstract characterization of operator spaces that allows us to define these spaces without a concrete representation. This result has had a tremendous impact and has led to the development of a general theory of operator spaces that parallels in some ways the development of the theory of Banach spaces.

Cited by 0

Hilbert space operator inequalities

D. J. H. Garling · Cambridge University Press eBooks · 2007

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

An operator space characterization of dual operator algebras

Christian Le Merdy · American Journal of Mathematics · 1999

This paper is devoted to dual algebras, that is w *-closed algebras of bounded operators on Hilbert space. We give abstract characterizations of unital dual algebras (resp. dual algebras) up to w *-homeomorphic unital complete isometry (resp. complete isomorphism). The results we obtain are natural w *-versions of analogous characterizations due to Blecher-Ruan-Sinclair and Blecher respectively, in the category of operator algebras.

Cited by 23

Abnormalities of an operator space

Krisko, Colin · Mspace (University of Manitoba) · 2024

A unital operator space 𝑋 is simply a subspace inside some C∗-algebra 𝒜 that contains the identity of 𝒜. Given a unital operator space 𝑋, a natural object of interest is the C∗-algebra generated by 𝑋. Notably, the structure of C∗(𝑋) is not determined by 𝑋 alone [Arv69]. To resolve this problem, one could instead consider the “smallest” C∗-algebra that can be generated by 𝑋. The C∗-envelope of 𝑋 is (if it exists) the C∗-algebra generated by a copy of 𝑋, that satisfies a particular universal property [Pau02]. An intimately related object is the Shilov ideal for 𝑋 which is (if it exists) the boundary ideal that contains every other boundary ideal [Arv69]. Here an ideal is called a boundary ideal when its quotient mapping restricts to a complete isometry on 𝑋 [Arv69]. The existence of these objects has attracted the attention of many: Notable contributions have been made by: Arveson [Arv69] [Arv08], Hamana [Ham79], Dritschel and McCullough [DM05], and Davidson and Kennedy [DK15]. More recently, Kakariadis [Kak13] presented a new proof of the existence. This proof yields a new characterization of the Shilov ideal as the restricted abnormalities of 𝑋. This thesis is devoted to the examination of the unrestricted abnormalities and studying what they can tell us about the operator space.

Cited by 0Open access

Operator Metric Space for Admissibility

Andrew John Paton · Zenodo (CERN European Organization for Nuclear Research) · 2026

This note introduces a metric structure over interpretive operators within the Paton System, defining quantitative distance to admissibility. The construct enables measurement of stability proximity, operator ordering effects, and boundary approach behaviour without modifying underlying equations. It formalises operator influence as admissibility distance shifts, establishing a universal comparison ruler for constraint-governed systems.

Cited by 0Open access

Exact Null-space Inverse Operator

Liam Covey · Zenodo (CERN European Organization for Nuclear Research) · 2026

The EEG inverse problem is fundamentally ill-posed due to the nontrivial null space of the forward operator mapping cortical activity to sensor measurements. Classical inverse methods address this ambiguity through regularization and prior assumptions, trading reconstruction accuracy for stability. In this work, we present an inverse model that operates directly in the null space of a fixed forward model. For a given forward operator, the proposed method reconstructs arbitrary source-space signals with error bounded by numerical precision. Reconstruction accuracy is invariant to additive sensor-space noise and requires negligible computational cost at inference time, enabling real-time application. We empirically validate exact reconstruction across diverse source priors and analyze failure modes under forward-model mismatch and null-space rank degradation. The results demonstrate that null-space inversion, when explicitly parameterized, admits exact and noise-robust reconstruction for fixed forward models. This work presents a self-contained demonstration of exact null-space inversion for fixed forward models.

Cited by 0Open access

Exact Null-space Inverse Operator

Liam Covey · Zenodo (CERN European Organization for Nuclear Research) · 2026

The EEG inverse problem is fundamentally ill-posed due to the nontrivial null space of the forward operator mapping cortical activity to sensor measurements. Classical inverse methods address this ambiguity through regularization and prior assumptions, trading reconstruction accuracy for stability. In this work, we present an inverse model that operates directly in the null space of a fixed forward model. For a given forward operator, the proposed method reconstructs arbitrary source-space signals with error bounded by numerical precision. Reconstruction accuracy is invariant to additive sensor-space noise and requires negligible computational cost at inference time, enabling real-time application. We empirically validate exact reconstruction across diverse source priors and analyze failure modes under forward-model mismatch and null-space rank degradation. The results demonstrate that null-space inversion, when explicitly parameterized, admits exact and noise-robust reconstruction for fixed forward models. This work presents a self-contained demonstration of exact null-space inversion for fixed forward models.

Cited by 0Open access

Exact Null-space Inverse Operator

Liam Covey · Zenodo (CERN European Organization for Nuclear Research) · 2026

The EEG inverse problem is fundamentally ill-posed due to the nontrivial null space of the forward operator mapping cortical activity to sensor measurements. Classical inverse methods address this ambiguity through regularization and prior assumptions, trading reconstruction accuracy for stability. In this work, we present an inverse model that operates directly in the null space of a fixed forward model. For a given forward operator, the proposed method reconstructs arbitrary source-space signals with error bounded by numerical precision. Reconstruction accuracy is invariant to additive sensor-space noise and requires negligible computational cost at inference time, enabling real-time application. We empirically validate exact reconstruction across diverse source priors and analyze failure modes under forward-model mismatch and null-space rank degradation. The results demonstrate that null-space inversion, when explicitly parameterized, admits exact and noise-robust reconstruction for fixed forward models. This work presents a self-contained demonstration of exact null-space inversion for fixed forward models.

Cited by 0Open access

Space‐variant aberrations: Differential operator

J. Ojeda‐Castañeda, J. Ibarra · Microwave and Optical Technology Letters · 1990

Abstract A differential operator describes the influence of space‐variant aberrations on a diffraction‐limited image, formed under coherent illumination. To illustrate our general treatment, the Talbot effect for space‐variant defocus is discussed in terms of the eigenfunctions of the differential operator.

Cited by 3

The Operator Hilbert Space OH . . .

Gilles Pisier · 2003

We prove that the operator Hilbert space OH does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge’s recent result that it admits such an embedding in the non semi-finite case.

Cited by 0