Limits Of Operators Research Articles

Universality of the Stochastic Airy Operator

AbstractWe introduce a new method for studying universality of random matrices. Let Tn be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, Tn converges to the stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, these ideas lead to conjectured operator limits for the entire family of soft edge distributions.© 2015 Wiley Periodicals, Inc.

Approximation of complex symmetric operators

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called a complex symmetric operator if there exists a conjugate-linear, isometric involution \(C:\mathcal {H}\rightarrow \mathcal {H}\) so that \(CTC=T^*\). In this paper, we study the approximation of complex symmetric operators. By virtue of an intensive analysis of compact operators in singly generated \(C^*\)-algebras, we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence. This gives a general solution to the norm closure problem for complex symmetric operators. As an application, we provide a concrete description of partial isometries which are norm limits of complex symmetric operators.

A 𝐶*-algebra approach to complex symmetric operators

In this paper, certain connections between complex symmetric operators and anti-automorphisms of singly generated C ∗ C^* -algebras are established. This provides a C ∗ C^* -algebra approach to the norm closure problem for complex symmetric operators. For T ∈ B ( H ) T\in \mathcal {B(H)} satisfying C ∗ ( T ) ∩ K ( H ) = { 0 } C^*(T)\cap \mathcal {K(H)}=\{0\} , we give several characterizations for T T to be a norm limit of complex symmetric operators. As applications, we give concrete characterizations for weighted shifts with nonzero weights to be norm limits of complex symmetric operators. In particular, we prove a conjecture of Garcia and Poore. On the other hand, it is proved that an essentially normal operator is a norm limit of complex symmetric operators if and only if it is complex symmetric. We obtain a canonical decomposition for essentially normal operators which are complex symmetric.

Inversion formulas for the Dunkl-type Segal–Bargmann transform

We give the inverse of the Dunkl-type Segal–Bargmann transform as a limit of certain integral operators. An application to Tikhonov regularization on the Dunkl-type Fock spaces is given using the theory of reproducing kernel Hilbert spaces.

Dense definiteness and boundedness of composition operators in L^2-spaces via inductive limits

The questions of dense definiteness and boundedness of composition operators in $L^2$-spaces are studied by means of inductive limits of operators. Methods based on projective systems of measure spaces and inductive limits of $L^2$-spaces are developed. Illustrative examples are presented.

Nonsymmetric difference Whittaker functions

Starting with nonsymmetric global difference spherical functions, we define and calculate spinor (nonsymmetric) global $$q$$ -Whittaker functions for arbitrary reduced root systems, which are reproducing kernels of the DAHA-Fourier transforms of Nil-DAHA and solutions of the $$q$$ -Toda–Dunkl eigenvalue problem. We introduce the spinor $$q$$ -Toda–Dunkl operators as limits of the difference Dunkl operators in DAHA theory under the spinor variant of the Ruijsenaars procedure. Their general algebraic theory (any reduced root systems) is the key part of this paper, based on the new technique of $$W$$ -spinors and corresponding developments in combinatorics of affine root systems.

Quantum quenches and generalized Gibbs ensemble in a Bethe Ansatz solvable lattice model of interacting bosons

We consider quantum quenches in the so-called q-boson lattice model. We argue that the Generalized Eigenstate Thermalization Hypothesis holds in this model, therefore the Generalized Gibbs Ensemble (GGE) gives a valid description of the stationary states in the long time limit. For a special class of initial states (which are the pure Fock states in the local basis) we are able to provide the GGE predictions for the resulting root densities. We also give predictions for the long-time limit of certain local operators. In the q → ∞ limit the calculations simplify considerably, the wave functions are given by Schur polynomials and the overlaps with the initial states can be written as simple determinants. In two cases we prove rigorously that the GGE prediction for the root density is correct. Moreover, we calculate the exact time dependence of a physical observable (the one-site Emptiness Formation Probability) for the quench starting from the state with exactly one particle per site. In the long-time limit the GGE prediction is recovered.

Normal limits of nilpotent operators in $C^*$-algebras

We will investigate the intersection of the normal operators with the closure of the nilpotent and quasinilpotent operators in various C*-algebras. A complete characterization will be given for type I and type III von Neumann algebras with separable predual and for unital, simple, purely infinite C*-algebras. Some restrictions that prevent normal operators from being limits of quasinilpotent operators will be given. The case of AF C*-algebras will be of particular interest. In addition, the closure of the span of the nilpotent operators and the distance from a projection to the nilpotent operators of various C*-algebras will be examined.

Limits of J-class operators

The purpose of the present work is to answer an open problem which is raised by G.Costakis and A.Manoussos in their paper oper- ators and hypercyclicity accepted by J. Operator Theory. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space.

A “Helium atom” of space: Dynamical instability of the isochoric pentahedron

We present an analysis of the dynamics of the equifacial pentahedron on the Kapovich-Millson phase space under a volume preserving Hamiltonian. The classical dynamics of polyhedra under such a Hamiltonian may arise from the classical limit of the node volume operators in loop quantum gravity. The pentahedron is the simplest nontrivial polyhedron for which the dynamics may be chaotic. We consider the distribution of polyhedral configurations throughout the space and find indications that the borders between certain configurations act as separatrices. We examine the local stability of trajectories within this phase space and find that locally unstable regions dominate although extended stable regions are present. Canonical and microcanonical estimates of the Kolmogorov-Sinai entropy suggest that the pentahedron is a strongly chaotic system. The presence of chaos is further suggested by calculations of intermediate time Lyapunov exponents which saturate to nonzero values.

Vortex Counting and Lagrangian 3-Manifolds

To every 3-manifold M one can associate a two-dimensional $${\mathcal{N}=(2, 2)}$$ supersymmetric field theory by compactifying five-dimensional $${\mathcal{N}=2}$$ super-Yang–Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional $${\mathcal{N}=2}$$ gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional $${\mathcal{N}=(2, 2)}$$ supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be mapped to the computation of degenerate conformal blocks in two-dimensional CFT’s. Degenerate limits of vertex operators in CFT receive a simple interpretation via geometric transitions in BPS counting.

Classical limit of the d-bar operators on quantum domains

We study one parameter families Dt, \documentclass[12pt]{minimal}\begin{document}$0<t<1$\end{document}0<t<1 of non-commutative analogs of the d-bar operator \documentclass[12pt]{minimal}\begin{document}$D_0 = \frac{\partial }{\partial \overline{z}}$\end{document}D0=∂∂z¯ on disks and annuli in complex plane and show that, under suitable conditions, they converge in the classical limit to their commutative counterpart. More precisely, we endow the corresponding families of Hilbert spaces with the structures of continuous fields over the interval [0, 1) and we show that the inverses of the operators Dt subject to APS boundary conditions form morphisms of those continuous fields of Hilbert spaces.

Product approximations for solutions to a class of evolution equations in Hilbert space

In this article we prove approximation formulae for a class of unitary evolution operators U(t,s)_{s,t\in [0,T] } associated with linear non-autonomous evolution equations of Schr\"{o}dinger type defined in a Hilbert space \mathcal{H} . An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for U(t,s) involve weak operator limits of products of suitable approximating functions taking values in \mathcal{L(H)} , the algebra of all linear bounded operators on \mathcal{H} . Our results may be relevant to the numerical analysis of U(t,s) and we illustrate them by considering two typical examples, including one related to the theory of time-dependent singular perturbations of self-adjoint operators.

Limits of kernel operators and the spectral regularity lemma

We study the spectral aspects of the graph limit theory. We give a description of graphon convergence in terms of convergence of eigenvalues and eigenspaces. Along these lines we prove a spectral version of Szemerédi’s regularity lemma. Using spectral methods we investigate group actions on graphons. As an application we show that the set of isometry invariant graphons on the sphere is closed in terms of graph convergence, however the analogous statement does not hold for the circle. This fact is rooted in the representation theory of the orthogonal group.

Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability

We review recent applications of the integrable discrete Hirota dynamics (Y-system) in the context of calculation of the planar AdS/CFT spectrum. We start from the description of solution of Hirota equations by the Bäcklund method where the requirement of analyticity results in the nested Bethe ansatz equations. Then, we discuss applications of the Hirota dynamics for the analysis of the asymptotic limit of long operators in the AdS/CFT Y-system.

Common knowledge and limit knowledge

We study the relationship between common knowledge and the sequence of iterated mutual knowledge from a topological point of view. It is shown that common knowledge is not equivalent to the limit of the sequence of iterated mutual knowledge. On that account the new epistemic operator limit knowledge is introduced and analyzed in the context of games. Indeed, an example is constructed where the behavioral implications of limit knowledge of rationality strictly refine those of common knowledge of rationality. More generally, it is then shown that limit knowledge of rationality is capable of characterizing any solution concept for some appropriate epistemic-topological conditions. Finally, some perspectives of a topologically enriched epistemic framework for games are discussed.

On wrapping corrections to GKP-like operators

In the recent paper arXiv:1010.5009, Maldacena et al. derive the two loop expressions for polygonal Wilson loops expectation values, or MHV amplitudes, by writing them as sums over exchanges of intermediate free particles. The spectrum of excitations of the flux tube between two null Wilson lines can be viewed as the spectrum of excitations around the infinite spin limit of finite twist operators in the sl(2) sector of N=4 SYM or the Gubser-Klebanov-Polyakov (GKP) string. This regime can be captured exploiting integrability and assuming that wrapping corrections are negligible compared to asymptotic Bethe Ansatz contributions. This assumption holds true for the N=4 SYM background GKP string, but deserves further analysis for excited states. Here, we investigate GKP cousins by considering various classes of (generalized) twist operators in beta-deformed N=4 SYM and ABJM theory. We show that the Y-system of Gromov-Kazakov-Vieira easily leads to accurate large spin expansions of the wrapping correction at lowest order in weak-coupling perturbation theory. As a byproduct, we confirm that wrapping corrections are subleading in all the considered cases.

Quantum singular operator limits of thin Dirichlet tubes via Γ-convergence

The $\Gamma$-convergence of lower bounded quadratic forms is used to study the singular operator limit of thin tubes (i.e., the vanishing of the cross section diameter) of the Laplace operator with Dirichlet boundary conditions; a procedure to obtain the effective Schr\"odinger operator (in different subspaces) is proposed, generalizing recent results in case of compact tubes. Finally, after scaling curvature and torsion the limit of a broken line is briefly investigated.

Beta ensembles, stochastic Airy spectrum, and a diffusion

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrödinger operator − d 2 d x 2 + x + 2 β b x ′ -\frac {d^2}{dx^2} + x + \frac {2}{\sqrt {\beta }} b_x^{\prime } restricted to the positive half-line, where b x ′ b_x^{\prime } is white noise. In doing so we extend the definition of the Tracy-Widom( β \beta ) distributions to all β > 0 \beta >0 and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converges to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

Prehospital induction techniques: mild hypothermia during advanced life support

The use of therapeutic hypothermia has been proposed as a potential treatment for a number of medical conditions including head injury and cerebral vascular accidents. This article aims to explore the different techniques used to induce hypothermia during prehospital cardiovascular arrest. The review of the necessary literature has been made in order to answer the question ‘What are the possible cooling methods to reach therapeutic hypothermia in out-of-hospital cardiac arrest?’. The research took place between May–November 2008. Ovid database was examined/questioned initially and the research was carried out simultaneously using Ovid Medline, Pubmed and CINAHL. The research showed that inclusion and exclusion criteria were similar in all the analysed studies. The chosen patients were aged 18–70 years. Only one article included patients older than 70 years who were affected by extra-hospital cardiocirculatory arrest and were treated with therapeutic hypothermia with infusion of cold liquids or use of correspondent remedies. A total of 627 620 articles were found during the search term research. Boolean operators were used to review the research. ‘AND’ was used to combine to the operator limit. ‘OR’ was used to include orthographic alternatives. All this reduced the research to four articles. Cutting off and mesh researches were used in ‘include relative terms’ within the Ovid control panel. The analysis of the four studies showed the effectiveness of introducing hypothermia in prehospital treatment dealing with the reduction of the oesophageal temperature. In all studies, the temperature reduction is relevant, presenting values of −1–4°C. The infusion of cold solutions is a technique that is easy to apply and has low costs. Infusion with cold pads, on the other hand, proves more difficult and requires greater attention and daily maintenance.