Showing papers on "Operator theory published in 2015"
TL;DR: Different methods that have been developed over the last decades to compute finite-dimensional approximations of infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - are reviewed.
Abstract: Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.
220 citations
TL;DR: In this paper, the pseudospectrum is given a central role in quantum mechanics with non-Hermitian operators, and the concept of pseudo-spectral properties is linked to quasi-hermiticity, similarity to self-adjoint operators and basis properties of eigenfunctions.
Abstract: We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.
172 citations
TL;DR: In this article, the authors discuss the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang-Baxter equation.
Abstract: We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain's ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin-1/2 chain, the Fermi-Hubbard chain, and the Lai-Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang-Baxter equation.
However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang-Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.
105 citations
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TL;DR: In this paper, the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas are surveyed, with a focus on analytic properties of Bloch and Fermi varieties.
Abstract: The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators.
The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential $-\Delta+V(x)$, but to a wide variety of elliptic periodic equations and systems, equations on graphs, $\overline{\partial}$-operator, and other operators on abelian coverings of compact bases.
Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.
91 citations
TL;DR: In this article, the authors studied the split common fixed point problem for two quasi-pseudo-contractive operators and proved strong convergence theorems for the problem and provided a unified framework for the study of this class problem and class of operators.
Abstract: The split common fixed point problem for two quasi-pseudo-contractive operators is studied. Some properties for quasi-pseudo-contractive operators are presented. An iterative algorithm for solving the split common fixed point problem for two quasi-pseudo-contractive operators is constructed. Strong convergence theorems are proved. A unified framework for the study of this class problem and class of operators is provided.
85 citations
TL;DR: In this paper, the authors explore an extension of Hilbert series techniques to count operators that include derivatives and find an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts.
Abstract: In this work, we explore an extension of Hilbert series techniques to count operators that include derivatives. For sufficiently low-derivative operators, we find an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts. Specifically, the technique can be applied whenever there is only one Lorentz invariant for a given partitioning of derivatives among the fields. At higher numbers of derivatives, equation of motion redundancies can be removed, but the increased number of Lorentz contractions spoils the subtraction of integration by parts redundancies. While restricted, this technique is sufficient to automatically generate the complete set of invariant operators of the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary numbers of flavors). At dimension 8, the algorithm does not automatically generate the complete operator set; however, it suffices for all but five classes of operators. For these remaining classes, there is a well defined procedure to manually determine the number of invariants. Using these methods, we thereby derive the set of 535 dimension-8 $N_f = 1$ operators.
81 citations
1 Dec 2015
TL;DR: It is discussed how the classical Hamilton-Jacobi-Bellman setting for optimal control can be reformulated in operator-theoretic terms and point the applicability of spectral operator theory in max-plus algebra to it.
Abstract: Recent contributions have extended the applicability of Koopman operator theory from dynamical systems to control. Stability theory got reformulated in terms of spectral properties of the Koopman operator [1], providing a nice link between the way we treat linear systems and nonlinear systems and opening the door for the use of classical linear e.g. pole placement theory in the fully nonlinear setting. New concepts such as isostables proved useful in the context of optimal control. Here, using Kato Decomposition we develop Koopman expansion for general LTI systems. We also interpret stable and unstable subspaces in terms of zero level sets of Koopman eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions to extend these results to globally stable systems. In conclusion, we discuss how the classical Hamilton-Jacobi-Bellman setting for optimal control can be reformulated in operator-theoretic terms and point the applicability of spectral operator theory in max-plus algebra to it. Geometric theories such as differential positivity have been also related to spectral theories of the Koopman operator [2], in cases when the attractor is a fixed point or a limit cycle, pointing the way to the more general case of quasiperiodic and chaotic attractors.
80 citations
TL;DR: In this article, the Frobenius norm of the commutator with the Hamiltonian has been shown to limit the time scale of relaxation of a spin-1/2 operator at high temperatures.
Abstract: We numerically construct slowly relaxing local operators in a nonintegrable spin-1/2 chain. Restricting the support of the operator to M consecutive spins along the chain, we exhaustively search for the operator that minimizes the Frobenius norm of the commutator with the Hamiltonian. We first show that the Frobenius norm bounds the time scale of relaxation of the operator at high temperatures. We find operators with significantly slower relaxation than the slowest simple "hydrodynamic" mode due to energy diffusion. Then we examine some properties of the nontrivial slow operators. Using both exhaustive search and tensor network techniques, we find similar slowly relaxing operators for a Floquet spin chain; this system is hydrodynamically "trivial," with no conservation laws restricting their dynamics. We argue that such slow relaxation may be a generic feature following from locality and unitarity.
76 citations
TL;DR: In this paper, the average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive-semidefinite part of the non-hermitians.
Abstract: In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, the reality of an eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators that may admit real eigenvalues under some symmetry conditions. In general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive-semidefinite part of the non-Hermitian operator. We also prove an uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under a quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula, and measuring the product of noncommuting projectors.
70 citations
TL;DR: The interpolation of multivariate data by operators of the neural network type is proved and a uniform approximation theorem with order is proved, showing these operators can also be used to approximate continuous functions defined on a box-domain of R(d).
64 citations
TL;DR: In this article, the linear positive operators generated by the q-Dunkl generalization of the exponential function are constructed and the rate of convergence of the operators for functions belonging to the Lipschitz class is presented.
Abstract: We construct the linear positive operators generated by the q-Dunkl generalization of the exponential function. We have approximation properties of the operators via a universal Korovkin-type theorem and a weighted Korovkin-type theorem. The rate of convergence of the operators for functions belonging to the Lipschitz class is presented. We obtain the rate of convergence by means of the classical, second order, and weighted modulus of continuity, respectively, as well.
TL;DR: In this article, the essential spectrum of the sum of two bounded linear operators defined on a Banach space is defined by means of the spectrum of each of the two operators, and a fine description of the Schechter essential spectrum for closed densely defined operators is given.
Abstract: In this paper, we present some results on Fredholm and upper semi-Fredholm operators involving demicompact operators. Our results generalize many known ones in the literature, in particular those obtained by Petryshyn in [27] and Jeribi et al. in [1], [22]. They are used to establish a fine description of the Schechter essential spectrum of closed densely defined operators, and to investigate the essential spectrum of the sum of two bounded linear operators defined on a Banach space by means of the essential spectrum of each of the two operators.
TL;DR: In this paper, the authors consider the notion of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties in pseudo-Hermitian quantum mechanics, and extend the analysis to operators in a partial inner product space, in particular the scale of Hilbert spaces generated by a single unbounded metric operator.
Abstract: Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties. Then we study quasi-Hermitian operators, bounded or not, that is, operators that are quasi-similar to their adjoint and we discuss their application in pseudo-Hermitian quantum mechanics. Finally, we extend the analysis to operators in a partial inner product space (\pip), in particular the scale of Hilbert spaces generated by a single unbounded metric operator.
TL;DR: In this article, the authors proved weighted $q$-variation inequalities with constant variance for differential and singular integral operators in higher dimensions, and the vector-valued extensions of these inequalities are also given.
Abstract: We prove weighted $q$-variation inequalities with $2
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TL;DR: In this paper, the authors prove Lieb-Thirring-type bounds for fractional Schrodinger operators and Dirac operators with complex-valued potentials, in the spirit of Frank and Sabin.
Abstract: We prove Lieb-Thirring-type bounds for fractional Schrodinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of Frank and Sabin.
TL;DR: In this article, the authors characterize the boundedness and compactness of linear operators induced by products of these operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball.
TL;DR: In this paper, it was shown that every 2-local derivation on a semi-finite von Neumann algebra is a derivation, and this result was extended to type III von NEumann algebras.
Abstract: The paper is devoted to the description of 2-local derivations on von Neumann algebras. Earlier it was proved that every 2-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type \(III\) von Neumann algebras. This implies that on arbitrary von Neumann algebra each 2-local derivation is a derivation.
TL;DR: The convergence in a norm of sequences generated by an iterative process for solving a variational inequality over the subset of fixed points of a quasi-nonexpansive operator $T$ defined on a Hilbert space is proved.
Abstract: In this paper, we prove the convergence in a norm of sequences generated by an iterative process for solving a variational inequality over the subset of fixed points of a quasi-nonexpansive operator $T$ defined on a Hilbert space. The process employs a sequence of quasi-nonexpansive operators for which the subset of common fixed points contains Fix$T$. We prove the convergence under a demi-closedness type condition for the sequence of operators as well as under the assumption that the process is approximately shrinking. We also give examples of methods satisfying these assumptions.
TL;DR: In this article, the authors show that the AGT correspondence to higher-rank class of supersymmetric gauge theories coincides with the skein algebra studied in the context of Chern-Simons theory.
Abstract: Non-perturbative aspects of $$ \mathcal{N}=2 $$
supersymmetric gauge theories of class $$ \mathcal{S} $$
are deeply encoded in the algebra of functions on the moduli space $$ {\mathrm{\mathcal{M}}}_{\mathrm{flat}} $$
of flat SL(N )- connections on Riemann surfaces. Expectation values of Wilson and ’t Hooft line operators are related to holonomies of flat connections, and expectation values of line operators in the low-energy effective theory are related to Fock-Goncharov coordinates on $$ {\mathrm{\mathcal{M}}}_{\mathrm{flat}} $$
. Via the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantization of Fock-Goncharov Laurent polynomials, and find that it coincides with the skein algebra studied in the context of Chern-Simons theory. Another realization of the skein algebra is generated by Verlinde network operators in Toda field theory. Comparing the spectra of these two realizations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalization of the AGT correspondence to higher-rank class $$ \mathcal{S} $$
theories.
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TL;DR: In this article, a survey of kernel analysis of Toeplitz operators is presented, starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.
Abstract: Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we will focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and P{\'o}lya sequences. Second, if the kernel is non-trivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.
TL;DR: In this paper, the Schatten class of Dirichlet realizations of non-self-adjoint differential operators with magnetic fields and complex electric potentials is defined. But the Schatz class is not defined in this paper.
Abstract: We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrodinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by Physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.
Journal Article•
TL;DR: In this article, it was shown that a commuting tuple (S, P ) with ∥S ∥ ≤ 2 and ∥P∥ ≤ 1 is a Γ-contraction if and only if there exists a bounded linear operator X such that S = X + X∗P and both X and X ∗ commutes with P and the numerical radius w(X) ≤ 1.
Abstract: A commuting pair of operators (S, P ) on a Hilbert space H is said to be a Γ-contraction if the symmetrized bidisc Γ = {(z1 + z2, z1z2) : |z1|, |z2| ≤ 1} is a spectral set of the tuple (S, P ). In this paper we develop some operator theory inspired by Agler and Young’s results on a model theory for Γ-contractions. We prove a Beurling-Lax-Halmos type theorem for Γ-isometries. Along the way we solve a problem in the classical one-variable operator theory, namely, a non-zero Mz-invariant subspace S of H E∗(D) is invariant under the analytic Toeplitz operator with the operatorvalued polynomial symbol p(z) = A + A∗z if and only if the Beurling-Lax-Halmos inner multiplier Θ of S satisfies (A+A∗z)Θ = Θ(B +B∗z), for some unique operator B. We use a ”pull back” technique to prove that a completely non-unitary Γ-contraction (S, P ) can be dilated to a pair (((A+AMz)⊕ U), (Mz ⊕Meit)), which is the direct sum of a Γ-isometry and a Γ-unitary on the Sz.-Nagy and Foias functional model of P , and that (S, P ) can be realized as a compression of the above pair in the functional model QP of P as (PQP ((A+A Mz)⊕ U)|QP , PQP (Mz ⊕Meit)|QP ). Moreover, we show that this representation is unique. We identify a complete set of unitary invariants for the class of completely non-unitary Γ-contractions. We prove that a commuting tuple (S, P ) with ∥S∥ ≤ 2 and ∥P∥ ≤ 1 is a Γ-contraction if and only if there exists a bounded linear operator X such that S = X +X∗P and both X and X∗ commutes with P and the numerical radius w(X) ≤ 1. In the commutant lifting set up, we obtain a unique and explicit solution to the lifting of S where (S, P ) is a completely non-unitary Γ-contraction. Our results concerning the Beurling-Lax-Halmos theorem of Γ-isometries and the functional model of Γ-contractions answers a pair of questions of J. Agler and N. J. Young.
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TL;DR: In this paper, the Stancu type generalization of Baskakov-Szasz operators based on the q-integers was considered and the rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function were also established for operators.
Abstract: In the present paper, we consider Stancu type generalization of Baskakov-Szasz operators based on the q-integers and obtain statistical and weighted statistical approximation properties of these operators. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established for operators.
TL;DR: A Chlodowsky type generalization of Szasz operators defined by means of the Brenke type polynomials is given, which studies the convergence of these operators in a weighted space of functions on a positive semi-axis.
Book•
15 May 2015
TL;DR: In this article, the authors present the spectral theorem as a statement on the existence of a unique continuous and measurable functional calculus, and present a proof without digressing into a course on the Gelfand theory of commutative Banach algebras.
Abstract: This book's principle goals are (i) to present the spectral theorem as a statement on the existence of a unique continuous and measurable functional calculus, (ii) to present a proof without digressing into a course on the Gelfand theory of commutative Banach algebras, (iii) to introduce the reader to the basic facts concerning the various von Neumann-Schatten ideals, the compact operators, the trace-class operators and all bounded operators, and finally, (iv) to serve as a primer on the theory of bounded linear operators on separable Hilbert space.
TL;DR: In this article, the perturbation of the generator is studied from the point of view of the generation of the evolution operator in the Riesz-Dunford functional calculus.
Abstract: The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator we study under which condition a closed operator P is such that T + P generates the evolution operator . This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.
TL;DR: In this paper, it was shown that the coefficients of abelian operators F n with MHV helicity configurations must satisfy a recursion relation, and are completely determined by that of F 4.
Abstract: We consider constraints on higher-dimensional operators for supersymmetric effective field theories. In four dimensions with maximal supersymmetry and SU(4) R-symmetry, we demonstrate that the coefficients of abelian operators F n
with MHV helicity configurations must satisfy a recursion relation, and are completely determined by that of F 4. As the F 4 coefficient is known to be one-loop exact, this allows us to derive exact coefficients for all such operators. We also argue that the results are consistent with the SL(2,Z) duality symmetry. Breaking SU(4) to Sp(4), in anticipation for the Coulomb branch effective action, we again find an infinite class of operators whose coefficients are determined exactly. We also consider three-dimensional $$ \mathcal{N} $$
= 8 as well as six-dimensional $$ \mathcal{N} $$
= (2,0),(1,0) and (1,1) theories. In all cases, we demonstrate that the coefficient of dimension-six operator must be proportional to the square of that of dimension-four.
TL;DR: A convergence theorem using the well known Bohman-Korovkin criterion is proven and the rate of convergence involving the modulus of continuity is established and a Voronovskaja type result is obtained for these operators.
TL;DR: In this paper, Ganikhodzhaev and Zanin generalized Zakharevich's example to the class of quadratic stochastic Volterra operators acting on a 2D simplex.
Abstract: On the basis of some numerical calculations, Ulam has conjectured that the ergodic theorem holds for any quadratic stochastic operator acting on a finite-dimensional simplex. However, Zakharevich showed that Ulam’s conjecture is false in general. Later, Ganikhodzhaev and Zanin generalized Zakharevich’s example to the class of quadratic stochastic Volterra operators acting on a 2D simplex. In this paper, we provide a class of nonergodic Lotka-Volterra operators which includes all previous operators used in this context.
TL;DR: In this article, the authors consider the problem of resolving operators of a fractional linear differential equation in a Banach space with a degenerate operator under the derivative and obtain necessary and sufficient conditions for relative p-boundedness of a pair of operators.
Abstract: We consider resolving operators of a fractional linear differential equation in a Banach space with a degenerate operator under the derivative. Under the assumption of relative p-boundedness of a pair of operators in this equation, we find the form of resolving operators and study their properties. It is shown that solution trajectories to the equation fill up a subspace of a Banach space. We obtain necessary and sufficient conditions for relative p-boundedness of a pair of operators in terms of families of resolving operators for degenerate fractional differential equation. Abstract results are illustrated by examples of the Cauchy problem for degenerate finite-dimensional system of fractional differential equations and of initial boundary-value problem for a fractional equation with respect to the time containing polynomials of Laplace operators with respect to spatial variables.