scholarly journals Otopy classification of gradient compact perturbations of identity in Hilbert space

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1677-1687
Author(s):  
Piotr Bartłomiejczyk ◽  
Piotr Nowak-Przygodzki
Keyword(s):  
Hilbert Space ◽  
Compact Perturbation ◽  
Compact Perturbations ◽  
Local Map ◽  
Perturbation Of Identity

We prove that the inclusion of the space of gradient local maps into the space of all local maps from Hilbert space to itself induces a bijection between the sets of the respective otopy classes of these maps, where by a local map we mean a compact perturbation of identity with a compact preimage of zero.

scholarly journals Global anomalies on the Hilbert space

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Diego Delmastro ◽  
Davide Gaiotto ◽  
Jaume Gomis
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Symmetry Algebra ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Strongly Coupled ◽  
Topological Quantum ◽  
Anomalous Theory

Abstract We show that certain global anomalies can be detected in an elementary fashion by analyzing the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert space are identified with the distinct cohomology “layers” that appear in the classification of anomalies in terms of cobordism groups. We illustrate the manifestation of the layers in the Hilbert for a variety of anomalous symmetries and spacetime dimensions, including time-reversal symmetry, and both in systems of fermions and in anomalous topological quantum field theories (TQFTs) in 2 + 1d. We argue that anomalies can imply an exact bose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrum of states; we provide a sharp characterization of when this phenomenon occurs and give nontrivial examples in various dimensions, including in strongly coupled QFTs. Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbert spaces, the action of operators and the modular data in spin TQFTs, material that can be read on its own.

On Compact Perturbations of Operators

1974 ◽  
Vol 26 (1) ◽  
pp. 247-250 ◽  
Author(s):  
Joel Anderson
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Unitary Operator ◽  
General Theorem ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Compact Perturbations ◽  
Small Norm

Recently R. G. Douglas showed [4] that if V is a nonunitary isometry and U is a unitary operator (both acting on a complex, separable, infinite dimensional Hilbert space ), then V — K is unitarily equivalent to V ⊕ U (acting on ⊕ ) where K is a compact operator of arbitrarily small norm. In this note we shall prove a much more general theorem which seems to indicate "why" Douglas' theorem holds (and which yields Douglas' theorem as a corollary).

Analysis and Classification of MoCap Data by Hilbert Space Embedding-Based Distance and Multikernel Learning

2019 ◽  
pp. 186-193
Author(s):  
Juan Diego Pulgarin-Giraldo ◽  
Andres Marino Alvarez-Meza ◽  
Steven Van Vaerenbergh ◽  
Ignacio Santamaría ◽  
German Castellanos-Dominguez
Keyword(s):  
Hilbert Space

The Classification of Factors is not Smooth

1973 ◽  
Vol 25 (1) ◽  
pp. 96-102 ◽  
Author(s):  
E. J. Woods
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Explicit Construction ◽  
Countable Family ◽  
Borel Sets ◽  
Isomorphism Classes ◽  
Borel Structure

There is a natural Borel structure on the set F of all factors on a separable Hilbert space [3]. Let denote the algebraic isomorphism classes in F together with the quotient Borel structure. Now that various non-denumerable families of mutually non-isomorphic factors are known to exist [1; 6; 8; 10; 11; 12; 13], the most obvious question to be resolved is whether or not is smooth (i.e. is there a countable family of Borel sets which separate points). We answer this question negatively by an explicit construction.

Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

2010 ◽  
Vol 62 (2) ◽  
pp. 305-319
Author(s):  
He Hua ◽  
Dong Yunbai ◽  
Guo Xianzhou
Keyword(s):  
Hilbert Space ◽  
Jacobson Radical ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Similarity Classification ◽  
Strongly Irreducible ◽  
Decomposable Operators ◽  
Decomposable Operator

AbstractLet 𝓗 be a complex separable Hilbert space and ℒ(𝓗) denote the collection of bounded linear operators on 𝓗. In this paper, we show that for any operator A ∈ ℒ(𝓗), there exists a stably finitely (SI) decomposable operator A∈, such that ‖A−A∈‖ 𝓗 ∈ andA′(A∈)/ rad A′(A∈) is commutative, where rad A′(A∈) is the Jacobson radical of A′(A∈). Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.

scholarly journals Yang-Baxter and the Boost: splitting the difference

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Marius de Leeuw ◽  
Chiara Paletta ◽  
Anton Pribytok ◽  
Ana L. Retore ◽  
Paul Ryan
Keyword(s):  
Hilbert Space ◽  
Sigma Model ◽  
Spin Chains ◽  
Baxter Equation ◽  
Regular Solutions ◽  
One Dimensional ◽  
Difference Form ◽  
The Difference ◽  
The One

In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times1616×16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2)𝔰𝔲(2)⊕𝔰𝔲(2) symmetry, which include the one-dimensional Hubbard model and the SS-matrix of the {AdS}_5 \times {S}^5AdS5×S5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

The Relationship Between Distance Formulae and Compact Perturbations for Reflexive Algebras

1990 ◽  
Vol 33 (4) ◽  
pp. 489-493 ◽  
Author(s):  
Kenneth R. Davidson
Keyword(s):  
Compact Perturbation ◽  
Nest Algebras ◽  
Completely Distributive ◽  
Compact Perturbations ◽  
Reflexive Algebras ◽  
Distance Formulae ◽  
The Relationship

AbstractFor completely distributive CSL algebras, hyper-reflexivity is equivalent to a description of the compact perturbation of the algebra analogous to the Fall-Arveson-Muhly Theorem for nest algebras.

scholarly journals Lie and Jordan structure in operator algebras

1980 ◽  
Vol 29 (2) ◽  
pp. 129-142
Author(s):  
Derek W. Robinson ◽  
Erling Størmer
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Jordan Algebra ◽  
Operator Algebras ◽  
Bounded Operators ◽  
Jordan Structure ◽  
Spectral Subspaces

AbstractLet υ be a C*-algebra, α a *-anti-automorphism of order 2, and υα(±1) = {A; A ∈ υ, α(A) = ± A} the spectral subspaces of α. It follows that υα(+ 1) is a Jordan algebra and υα(− 1) is a Lie algebra. We begin the classification of pairs of Jordan and Lie algebras which can occur in this manner by examining υ = ℒ(ℋ), the algebra of bounded operators on a Hilbert space ℋ.

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