Matrix Representation Of Operator Research Articles

Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} (d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document}), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

Matrix representation of Toeplitz operators on Newton spaces

In this paper, we study several properties of an orthonormal basis {Nn(z)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{N_{n}(z)\\}$\\end{document} for the Newton space N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}({\\mathbb{P}})$\\end{document}. In particular, we investigate the product of Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N_{m}$\\end{document} and Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N_{m}$\\end{document} and the orthogonal projection P of Nn‾Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline{N_{n}}N_{m}$\\end{document} that maps from L2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}(\\mathbb{P})$\\end{document} onto N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}(\\mathbb{P})$\\end{document}. Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}({\\mathbb{P}})$\\end{document}.

Quantum α-fidelity of diagonalizable second-order Hamiltonians in multi-dimensional systems

How to characterize the similarity between two quantum states has received much attention recently. Motivated by the continued push in this direction, we propose a computable analytical formula for the quantum α-fidelity between diagonalizable second-order Hamiltonians, which is simply expressed in terms of the matrix representation of operators. Further, using this analytical formula, we calculate the α-fidelity of the one-dimensional and two-dimensional displaced-squeezed thermal states, the Liu-Tombesi oscillators, and twin-beam amplitude-squeezed states. Since α-fidelity can precisely characterize the approximation accuracy between two quantum states by adjusting the value of the parameter α, our results offer a richer fidelity family to quantify the similarity between diagonalizable second-order Hamiltonians in multi-dimensional systems, and hence may have potential applications in quantum information and quantum computing.

SQMBox: Interfacing a semiempirical integral library to modular ab initio electronic structure enables new semiempirical methods.

Ab initio and semiempirical electronic structure methods are usually implemented in separate software packages or use entirely different code paths. As a result, it can be time-consuming to transfer an established ab initio electronic structure scheme to a semiempirical Hamiltonian. We present an approach to unify ab initio and semiempirical electronic structure code paths based on a separation of the wavefunction ansatz and the needed matrix representations of operators. With this separation, the Hamiltonian can refer to either an ab initio or semiempirical treatment of the resulting integrals. We built a semiempirical integral library and interfaced it to the GPU-accelerated electronic structure code TeraChem. Equivalency between ab initio and semiempirical tight-binding Hamiltonian terms is assigned according to their dependence on the one-electron density matrix. The new library provides semiempirical equivalents of the Hamiltonian matrix and gradient intermediates, corresponding to those provided by the ab initio integral library. This enables the straightforward combination of semiempirical Hamiltonians with the full pre-existing ground and excited state functionality of the ab initio electronic structure code. We demonstrate the capability of this approach by combining the extended tight-binding method GFN1-xTB with both spin-restricted ensemble-referenced Kohn-Sham and complete active space methods. We also present a highly efficient GPU implementation of the semiempirical Mulliken-approximated Fock exchange. The additional computational cost for this term becomes negligible even on consumer-grade GPUs, enabling Mulliken-approximated exchange in tight-binding methods for essentially no additional cost.

Wavelets based computational algorithms for multidimensional distributed order fractional differential equations with nonlinear source term

In this manuscript we provide effective computational algorithms based on Legendre wavelet (LW), Bernstein wavelet (BW), and standard tau approach to approximate the solution of viscoelasticity damping motion nonlinear distributed order fractional differential equation (VDMNDOFDE) and multi-dimensional distributed order time-fractional nonlinear partial differential equation (DOT-FNPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving VDMNDOFDE and DOT-FNPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and product term associated with the integral based on LW and BW are established to find the numerical solutions of the proposed VDMNDOFDE and DOT-FNPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish VDMNDOFDE and DOT-FNPDE into system of nonlinear algebraic equations. Here, these systems of nonlinear algebraic equations are solved by the Newton's method to get the value of unknown vectors. Error bounds, convergence analysis, numerical algorithms of the VDMNDOFDE and DOT-FNPDE are rigorously investigated. For the reliability of the proposed computational algorithm, numerous test examples have been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique. Also, the proposed technique found to be more accurate with the existing scheme.

Some results on higher order symmetric operators

For some operator A ? B(H), positive integers m and k, an operator T ? B(H) is called k-quasi-(A,m)-symmetric if T*k( mP j=0 (?1)j(m j )T*m?jATj)Tk = 0, which is a generalization of the m-symmetric operator. In this paper, some basic structural properties of k-quasi-(A,m)-symmetric operators are established with the help of operator matrix representation. We also show that if T and Q are commuting operators, T is k-quasi-(A,m)-symmetric and Q is n-nilpotent, then T + Q is (k + n ? 1)-quasi-(A,m + 2n ? 2)-symmetric. In addition, we obtain that every power of k-quasi-(A,m)-symmetric is also k-quasi-(A,m)-symmetric. Finally, some spectral properties of k-quasi-(A,m)-symmetric are investigated.

Extension of m-Symmetric Hilbert Space Operators

We introduce a new class of operators, which we will call the class ofP-quasi-m-symmetric operators that includesm-symmetric operators andk-quasim-symmetric operators. Some basic structural properties of this class of operators are established based on the operator matrix representation associated with such operators.

Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction

Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such macroscopically LIOMs, we explore and study some of their structural properties across the MBL transition. We first, consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a non-perturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert-space. Our cluster size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert-space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\nu$ both in the presence and absence of interaction. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.

Operators with Brownian unitary dilations

Certain bounded linear operators $T$ on a complex Hilbert space $\h$ which have 2-isometric liftings $S$ on another space $\ka \supset \h$ are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition $S^*S\h \subset \h$. Furthermore we describe other types of dilations for $T$, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators.

The Determination of Eigenvalues and Eigenvectors of the Orbital Angular Momentum

The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.

Asymptotical Stabilization for Probabilistic Boolean Control Networks under Operators and Inputs Constraints

In this paper, asymptotical stabilization of probabilistic Boolean control networks (<monospace>PBCNs</monospace>) with a switching signal being an independent identically distributed process has been systematically studied. Here, the binary operators connecting logical functions of state nodes and pinning controllers are constrained to certain sets. By establishing matrix representation of operators and input constraints, several criteria for asymptotical stabilization have been established. The proposed asymptotical stabilization extends the results of Boolean networks (<monospace>BNs</monospace>) under static pinning control, which is more practical in real-word control design systems. Finally, simulations are given to verify the effectiveness of the obtained theoretical results.

Dual finite frames for vector spaces over an arbitrary field with applications

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

On the class of $k$-quasi-$(n,m)$-power normal operators

We introduce a family of operators called the family of $k$-quasi-$(n,m)$-power normal operators. Such family includes normal, $n$-normal and $(n,m)$-power normal operators. An operator $T \in {\mathcal B}({\mathcal H})$ is said to be $k$-quasi-$(n,m)$-power normal if it satisfies $$T^{*k}\bigg(T^nT^{*m}-T^{*m}T^n\bigg)T^k=0,$$ where $k,n$ and $m$ are natural numbers. Firstly, some basic structural properties of this family of operators are established with the help of special kind of operator matrix representation associated with such family of operators. Secondly, some properties of\linebreak algebraically $k$-quasi-$(n,m)$-power normal operators are discussed. Thirdly, we consider the study of tensor products of $k$-quasi-$(n,m)$-power normal operators. A necessary and sufficient condition for $T\otimes S$ to be a $k$-quasi-$(n,m)$-power normal is given, when $T \neq0$ and $S\neq0$.

A note on multigrid preconditioning for fractional PDE-constrained optimization problems

In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order optimality Karush–Kuhn–Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the state constraint using the control-to-state map. Our multigrid preconditioning approach shows a dramatic reduction in the number of CG iterations. We assess the quality of preconditioner in terms of the spectral distance. Finally, we provide a partial theoretical analysis for this preconditioner, and we formulate a conjecture which is clearly supported by our numerical experiments.

Generalized Jacobi Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients

This paper is devoted to the matrix representation of ordinary symmetric differential operators with polynomial coefficients on the whole axis. We prove that in this case, generalized Jacobi matrices appear. We examine the problem of defect indexes for ordinary differential operators and generalized Jacobi matrices corresponding to them in the spaces L2(−∞,+∞) and l2, respectively, and analyze the spectra of self-adjoint extensions of these operators (if they exist). This method allows one to detect new classes of entire differential operators of minimal type (in the sense of M. G. Krein) with certain defect numbers. In this case, the defect numbers of these operators can be not only less than or equal, but also greater than the order of the corresponding differential expressions. In particular, we construct examples of entire differential operators of minimal type that are generated by irregular differential expressions.

Operator matrix representations of the Generalised Bott–Duffin Inverse

In this paper, we give the operator matrix representation of the generalised Bott–Duffin inverse $$A_{(\mathcal {M},\mathcal {N})}^{\dagger }= P_{\mathcal {M},\mathcal {N}} \left( AP_{\mathcal {M},\mathcal {N}} + P_{\mathcal {N},\mathcal {M}}\right) ^\dagger $$ using results that concern lower-triangular operator matrices. Specially, we extend some previous results and we give necessary and sufficient conditions for $$A_{(\mathcal {M},\mathcal {N})}^\dagger = (P_{\mathcal {M},\mathcal {N}}AP_{\mathcal {M},\mathcal {N}})^\dagger $$ to hold.

Spectral properties and the dynamics of quasi-2-expansive operators

An operator T on a complex Hilbert space is called a quasi-2-expansive operator if T^{\*3}T^{3}-2T^{\*2}T^{2}+T^{\*}T\leq0, which is a further generalization of 2-isometric operator. In this paper, some basic structural properties of quasi-2-expansive operators are established with the help of operator matrix representation. Using these results, we show that the spectrum is continuous on the class of all quasi-2-expansive operators. Moreover, we verify that Weyl's theorem holds for quasi-2-expansive operators. Finally, we prove that a weakly supercyclic quasi-2-expansive operator is a unitary operator.

On n-quasi-[m,C]-isometric operators

For positive integers m and n, an operator T in B ( H ) is said to be an n-quasi-[m,C]-isometric operator if there exists some conjugation C such that . In this paper, some basic structural properties of n-quasi-[m,C]-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi-[m,C]-isometric operator is again an n-quasi-[m,C]-isometric operator. Moreover, we show that the class of n-quasi-[m,C]-isometric operators is norm closed. Finally, we examine the stability of n-quasi-[m,C]-isometric operator under perturbation by nilpotent operators commuting with T.

Matrix representations of multidimensional integral and ergodic operators

We provide a representation of the C*-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the explicit functional calculus on this algebra. The method can be useful in various applications, since many discrete approximations of integral and differential operators belong to this algebra. Some examples are also presented: (1) we construct an explicit functional calculus for extended Fredholm integral operators with piecewise constant kernels, (2) we find a wave function and spectral estimates for 3D discrete Schrödinger equation with planar, guided, local potential defects, and point sources. The accuracy of approximation of continuous multi-kernel integral operators by the operators with piecewise constant kernels is also discussed.

G-frames and their generalized multipliers in Hilbert spaces

In this article, we introduce the concept of generalized multipliers for g-frames. In fact, we show that every generalized multiplier for g-Bessel sequences is a generalized multiplier for the induced sequences, in a special sense. We provide some sufficient and/or necessary conditions for the invertibility of generalized multipliers. In particular, we characterize g-Riesz bases by invertible multipliers. We look at which perturbations of g-Bessel sequences preserve the invertibility of generalized multipliers. Finally, we investigate how to find a matrix representation of operators on a Hilbert space using g-frames, and then we characterize g-Riesz bases and g-orthonormal bases by applying such matrices.