Identity Operator Research Articles

A Rigorous Algorithm for the Boundary Value Problem of Dielectric Bodies Based on the Analytical Regularization Method

In this paper, the application of the Analytical Regularization Method (ARM) to a dielectric cylinder is presented. At first, the matrix of the algebraic system is constructed as a 2 × 2 block matrix where the off-diagonal blocks consist of the Fourier coefficients of non-singular operators and the entries of the diagonal blocks involve the associated singularities. Then the left- and right-hand side operators of the ARM built up on these singularities are applied to the algebraic system, transforming it to a second kind one in form of $(I + H)y=b$ where $I$ the identity operator and $H$ is a compact operator in space $l_{2}$ . The algorithm is applied to a dielectric cylinder illuminated by a TM -z polarized plane wave. The formulation of TE-z polarization is almost the same except the field functions will be in terms of magnetic field and its normal derivative. The numerical results are validated by the solution of separation of variables for a circular dielectric cylinder and then the superiority of the second kind system to the first kind one is shown by means of the condition number with respect to the truncation number of the algebraic system.

Quantum channels transforming Bell local states as unsteerable states

Motivated by the contributions [Sci. Rep. 6, 39063 (2016)] and [Phys. Rev. A 98, 052114 (2018)], we aim to establish a new method for theoretically checking Bell nonlocality of a bipartite state of systems for any finite dimensional Hilbert spaces and , provided that the operator space has an orthonormal basis consisting of the identity operator and Hermitian unitary operators. Our main result induces a quantum channel transforming a Bell local state as an unsteerable state from A to B whenever the parameters T and satisfy the condition . Thus, when and the image is steerable from Ato B, the original state must be Bell nonlocal. Our result not only reveals a deep connection between EPR steerability and Bell nonlocality but also provides a feasible approach to theoretically prove difficultly verified Bell nonlocality by translating it into easily verified steerability.

Space-time fractional diffusion equation associated with Jacobi expansions

We consider the operator , where is the three term recurrence relation for the Jacobi polynomial normalized at the point x = 1 and I is the identity operator acting on . Knowing that for , these polynomials satisfy a product formula which provides a hypergroup structure on , we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation Here, , is the Caputo time fractional derivative of order σ. The solution is given by , being the fundamental solution expressed in terms of Mittag-Leffler function and is the convolution on the hypergroup . We prove that, for , is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution.

Generalized discrete truncated Wigner approximation for nonadiabatic quantum-classical dynamics.

Nonadiabatic molecular dynamics occur in a wide range of chemical reactions and femtochemistry experiments involving electronically excited states. These dynamics are hard to treat numerically as the system's complexity increases, and it is thus desirable to have accurate yet affordable methods for their simulation. Here, we introduce a linearized semiclassical method, the generalized discrete truncated Wigner approximation (GDTWA), which is well-established in the context of quantum spin lattice systems, into the arena of chemical nonadiabatic systems. In contrast to traditional continuous mapping approaches, e.g., the Meyer-Miller-Stock-Thoss and the spin mappings, GDTWA samples the electron degrees of freedom in a discrete phase space and thus forbids an unphysical unbounded growth of electronic state populations. The discrete sampling also accounts for an effective reduced but non-vanishing zero-point energy without an explicit parameter, which makes it possible to treat the identity operator and other operators on an equal footing. As numerical benchmarks on two linear vibronic coupling models and Tully's models show, GDTWA has a satisfactory accuracy in a wide parameter regime, independent of whether the dynamics is dominated by relaxation or by coherent interactions. Our results suggest that the method can be very adequate to treat challenging nonadiabatic dynamics problems in chemistry and related fields.

Extended dynamic mode decomposition for inhomogeneous problems

Dynamic mode decomposition (DMD) is a powerful data-driven technique for construction of reduced-order models of complex dynamical systems. Multiple numerical tests have demonstrated the accuracy and efficiency of DMD, but mostly for systems described by homogeneous partial differential equations (PDEs) with homogeneous boundary conditions. We propose an extended dynamic mode decomposition (xDMD) approach to cope with the potential unknown sources/sinks in PDEs. Motivated by similar ideas in deep neural networks, we equip our xDMD with two new features. First, it has a bias term, which accounts for inhomogeneity of PDEs and/or boundary conditions. Second, instead of learning a flow map, xDMD learns the residual increment by subtracting the identity operator. Our theoretical error analysis demonstrates the improved accuracy of xDMD relative to standard DMD. Several numerical examples are presented to illustrate this result.

Small operator ideals on the Schlumprecht and Schreier spaces

We present a criterion for operators on a Banach space X to generate distinct operator ideals in the algebra B(X) of bounded linear operators on X. We show that there are exactly 2c distinct small closed operator ideals on the Schlumprecht space and on any Schreier space of finite order.

О некоторых числах оператора Харди в пространствах Лоренца

In the paper conditions are found under which the compact operator $Tf(x)=\varphi(x)\int_0^ xf(\tau)v(\tau)\,d\tau,$ $x>0,$ acting in weighted Lorentz spaces $T:L^{r,s}_{v} (\mathbb{R^+})\to L^{p,q}_{\omega}(\mathbb{R^+})$ in the domain $1<\max (r,s)\le \min(p,q)<\infty,$ belongs to operator ideals $\mathfrak{S}^{(a)}_\alpha$ and $\mathfrak{E}_\alpha$, $0<\alpha<\infty$. And estimates are also obtained for the quasinorms of operator ideals in terms of integral expressions which depend on operator weight functions.

Identical approximation operator and regularization method for the Cauchy problem of 2‐D heat conduction equation

The purpose of this work is to create an identical approximate regularization method for solving a Cauchy problem of two‐dimensional heat conduction equation. The problem is severely ill‐posed. The convergence rates are obtained under a priori regularization parameter choice rule. Numerical results are presented for two examples with smooth and continuous but not smooth boundaries and compared the identical approximate regularization solutions which are displayed in text. The numerical results show that our method is effective, accurate, and stable to solve the ill‐posed Cauchy problems.

Ideals of hypercyclic operators that factor through $\ell ^p$

We study the injective and surjective hull of operator ideals generated by hypercyclic backward weighted shifts that factor through $\ell^p$.

Ergodic theorems in Banach ideals of compact operators

Ergodic theorems in Banach ideals of compact operators

Aggregation of indistinguishability operators

The paper studies the aggregation of pairs of T-indistinguishability operators. More concretely we address the question whether A(E,E′) is a T-indistinguishability operator if E,E′ are T-indistinguishability operators. The answer depends on the aggregation function, the t-norm T, and the chosen T-indistinguishability operators. It is well-known that an aggregation function preserves T-transitive relations if and only if it dominates the t-norm T. We show the important role of the minimum t-norm TM in this preservation problem. In particular we develop weaker forms of domination that are used to provide characterizations of TM-indistinguishability preservation under aggregation. We also prove that the existence of a single strictly monotone aggregation that satisfies the indistinguishability operator preservation property guarantees all aggregations to have the same preservation property.

Probability Error Bounds for Approximation of Functions in Reproducing Kernel Hilbert Spaces

We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X , firstly in terms of finite linear combinations of functions of type K x i and then in terms of the projection π x n on span K x i i = 1 n , for random sequences of points x = x i i in X . Given a probability measure P , letting P K be the measure defined by d P K x = K x , x d P x , x ∈ X , our approach is based on the nonexpansive operator L 2 X ; P K ∋ λ ↦ L P , K λ ≔ ∫ X λ x K x d P x ∈ H , where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by H P , that is the operator range of L P , K . Our main result establishes bounds, in terms of the operator L P , K , on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type K x i , for x i i sampled independently from P , falls below a given threshold. For sequences of points x i i = 1 ∞ constituting a so-called uniqueness set, the orthogonal projections π x n to span K x i i = 1 n converge in the strong operator topology to the identity operator. We prove that, under the assumption that H P is dense in H , any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or L p norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H 2 D are presented as well.

Learning an Optimal Feedback Operator Semiglobally Stabilizing Semilinear Parabolic Equations

Stabilizing feedback operators are presented which depend only on the orthogonal projection of the state onto the finite-dimensional control space. A class of monotone feedback operators mapping the finite-dimensional control space into itself is considered. The special case of the scaled identity operator is included. Conditions are given on the set of actuators and on the magnitude of the monotonicity, which guarantee the semiglobal stabilizing property of the feedback for a class of semilinear parabolic-like equations. Subsequently an optimal feedback control minimizing the quadratic energy cost is computed by a deep neural network, exploiting the fact that the feedback depends only on a finite dimensional component of the state. Numerical simulations demonstrate the stabilizing performance of explicitly scaled orthogonal projection feedbacks, and of deep neural network feedbacks.

Ideals of band-dominated operators

Based on limit operator techniques, we introduce and study an increasing family of closed ideals of the algebra of the band-dominated operators on ), also known as the uniform Roe algebra of the coarse space . We describe the centers of the quotients of by these ideals and identify them with higher-order slowly oscillating functions.

Schatten classes for Hilbert modules over commutative C⁎-algebras

We define Schatten classes of adjointable operators on Hilbert modules over abelian C⁎-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a C⁎-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm–continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the C⁎-valued Fredholm determinant and operator zeta functions, and propose a notion of p-summable unbounded Kasparov cycles in the commutative setting.

On functional equations related to generalized inner derivations on standard operator algebras

Our aim is to prove the following result. Let X be a real or complex Banach space, let ℒ(X) be the algebra of all bounded linear operators on X and let 𝒜(X)⊆ℒ(X) be a standard operator algebra, which possesses the identity operator. Suppose there exists a linear mapping F:𝒜(X)→ℒ(X) satisfying the relation F(An+2)= ∑i=1n+1Ai−1F(A2)An+1−i−∑i=1nAiF(A)An+1−i for all A∈𝒜(X) and some fixed integer n≥1. In this case F is of the form F(A)=AB1+B2A for all A∈𝒜(X) and some fixed B1,B2∈ℒ(X). In particular, F is continuous.

Multiplicative *-Lie type higher derivations of standard operator algebras

Let be a standard operator algebra on an infinite dimensional complex Hilbert space containing identity operator I. Let be the polynomial defined by n indeterminates and their multiple *-Lie products and be the set of non-negative integers. In this paper, it is shown that if is closed under the adjoint operation and is the family of mappings such that the identity map on satisfying for all and for each , then is an additive *-higher derivation. Moreover, is inner.

Ergodic theorems for d-dimensional flows in ideals of compact operators

Ergodic theorems for d-dimensional flows in ideals of compact operators

Numerical Index and Daugavet Property of Operator Ideals and Tensor Products

We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain space and the range space. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain space, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor product of two Banach spaces has the Daugavet property and the unit ball of one of the factor is slicely countably determined or its dual contains a point of Fréchet differentiability of the norm, then the other factor inherits the Daugavet property. If an injective tensor product of two Banach spaces has the Daugavet property and one of the factors contains a point of Fréchet differentiability of the norm, then the other factor has the Daugavet property.

Void formation in operator growth, entanglement, and unitarity

The structure of the Heisenberg evolution of operators plays a key role in explaining diverse processes in quantum many-body systems. In this paper, we discuss a new universal feature of operator evolution: an operator can develop a void during its evolution, where its nontrivial parts become separated by a region of identity operators. Such processes are present in both integrable and chaotic systems, and are required by unitarity. We show that void formation has important implications for unitarity of entanglement growth and generation of mutual information and multipartite entanglement. We study explicitly the probability distributions of void formation in a number of unitary circuit models, and conjecture that in a quantum chaotic system the distribution is given by the one we find in random unitary circuits, which we refer to as the random void distribution. We also show that random unitary circuits lead to the same pattern of entanglement growth for multiple intervals as in (1 + 1)-dimensional holographic CFTs after a global quench, which can be used to argue that the random void distribution leads to maximal entanglement growth.