Singular Operators Research Articles

Properties of the resolvent of singular q-Dirac operators

In this article, we investigate the resolvent operator of a singular q-Dirac system. We obtain an integral representations for the resolvent of this system, in terms of the spectral function. Furthermore, we give a formula for the Titchmarsh-Weyl function of q-Dirac system using the integral representation of the resolvent. For more information see: https://ejde.math.txstate.edu/Volumes/2020/03/abstr.html

One-dimensional perturbations of unitaries that are quasiaffine transforms of singular unitaries, and multipliers between model spaces

It is shown that, under some natural additional conditions, an operator which intertwines one cyclic singular unitary operator with one dimensional perturbation of another cyclic singular unitary operator is the operator of multiplication by a multiplier between model spaces. Using this result, it is shown that if $T$ is one dimensional perturbation of a unitary operator, $T$ is a quasiaffine transform of a singular unitary operator, and $T$ is power bounded, then $T$ is similar to a unitary operator, and $\sup_{n\geq 0}\|T^{-n}\|\leq(2(\sup_{n\geq 0}\|T^n\|)^2+1)\cdot(\sup_{n\geq 0}\|T^n\|)^5$.

Optimal Operator Preconditioning for Galerkin Boundary Element Methods on 3-Dimensional Screens

We consider first-kind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in $\mathbb{R}^{3}$ and their Galerkin discretization by means of low-order piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calderón-type preconditioners based on (i) new boundary integral operators, which provide the exact inverses of the weakly singular and hypersingular operators on flat disks, and (ii) stable duality pairings relying on dual meshes. On screens obtained as images of the unit disk under bi-Lipschitz transformations, this approach achieves condition numbers uniformly bounded in the meshwidth even on locally refined meshes. Comprehensive numerical tests also confirm its excellent preasymptotic performance.

Explicit Krein resolvent identities for singular Sturm-Liouville operators with applications to Bessel operators

We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+(\nu^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $\nu\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.

On the Cauchy Problem for the Iterated Generalized Two-axially Symmetric Equation of Hyperbolic Type

The Cauchy problem for an iterated generalized two-axially symmetric equation of hyperbolic type is investigated. Unlike traditional methods, the generalized Erdelyi–Kober operator of fractional order is applied to solve the problem. The application of this operator allows us to reduce the equations with the lowest term and with the singular Bessel operator acting in one of the variables to an equation without the Bessel operator acting on this variable and without the lowest term. The proposed approach makes it possible to construct an explicit formula for solving the formulated problem. These formulas express solutions of the Cauchy problem in terms of the sum of the initial functions under the action of powers of the Bessel operator. They allow us to directly see the character of the dependence of the solution on the initial functions.

On triangle inequality for Miranda-Thompson’s majorization and gradients of increasing functions

In this paper, we establish some refinements of the following triangle inequality proved by T.-Y. Tam: $$ {{\widetilde{\sigma }}} (x + y) \prec _{mt} {{\widetilde{\sigma }}} (x) + {{\widetilde{\sigma }}} (y) $$ for real matrices x and y with the modified singular value operator $$ {{\widetilde{\sigma }}} (\cdot ) $$ and Miranda-Thompson’s majorization $$ \prec _{mt} $$ on $$ {\mathbb {R}}^{n}$$ related to the group of special orthogonal matrices.

A half-inverse problem for the singular diffusion operator with jump conditions

In this paper, half inverse spectral problem for diffusion operator with jump conditions dependent on the spectral parameter and discontinuoty coeffcient is considered. The half inverse problems is studied of determining the coeffcient and two potential functions of the boundary value problem its spectrum by Hocstadt- Lieberman and Yang-Zettl methods. We show that two potential functions on the whole interval and the parameters in the boundary and jump conditions can be determined from spectrum.

Multilevel Schwarz preconditioners for singularly perturbed symmetric reaction-diffusion systems

We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.

Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

General Euler-Poisson-Darboux Equation and Hyperbolic B-Potentials

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz B-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler-Poisson-Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

Regular approximation of linear Hamiltonian operators with two singular endpoints

In this paper we investigate regular approximation of linear Hamiltonian operators with two singular endpoints. For any given singular self-adjoint Hamiltonian operator, its spectrum can be approximated with the eigenvalues of a sequence of regular operators. The inherited restriction operators are constructed by the limit-circle solutions and the limit-point solutions, and proved to be self-adjoint. Via the abstract operator theory, we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum. Moreover for the case with both endpoints being limit circle, spectral exactness of spectrum is obtained by Hilbert-Schmidt norm convergence. In the case of Friedrichs extension of singular Hamiltonian systems, spectral exactness also holds due to variational principle of eigenvalues.

Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

We discuss the spectral properties of singular Schrodinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case, the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maximized when the arms are arranged in one of the known five sharp configurations.

Complete transformation of Radon-Kipriyanov. Some properties

The even Radon-Kipriyanov transform (Kg-transform) is suitable for investigating problems with the Bessel singular differential operator Bi = 2i2+iii,i 0. In this paper, we introduce the odd Radon-Kipriyanov transform and complete Radon-Kipriyanov transform to investigation more general equations containing odd B‑derivativesiBik, k = 0, 1, 2, ... (in particular, gradients of functions). Formulas of K-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B. M. Levitan and the odd Bessel transform introduced by I. A. Kipriyanov and V. V. Katrakhov, a connection was obtained between the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform. An analogue of Helgasons support theorem and an analogue of the Paley-Wiener theorem are presented.

Complete Radon–Kipriyanov Transform: Some Properties

The even Radon–Kipriyanov transform (Kγ-transform) is suitable for studying problems with the Bessel singular differential operator $${{B}_{{{{\gamma }_{i}}}}} = \frac{{{{\partial }^{2}}}}{{\partial x_{i}^{2}}} + \frac{{{{\gamma }_{i}}}}{{{{x}_{i}}~}}\frac{\partial }{{\partial {{x}_{i}}}},{{\gamma }_{i}} > 0$$. In this work, the odd Radon–Kipriyanov transform and the complete Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives $$\frac{\partial }{{\partial {{x}_{i}}}}~B_{{{{\gamma }_{i}}}}^{k},~~k = 0, 1, 2,~ \ldots $$ (in particular, gradients of functions). Formulas of the Kγ-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B.M. Levitan and the odd Bessel transform introduced by I.A. Kipriyanov and V.V. Katrakhov, a relationship of the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform is deduced. An analogue of Helgason’s support theorem and an analog of the Paley–Wiener theorem are given.

Convergence of Spectral Decompositions for a Singular Differential Operator with General Boundary Conditions

We investigate the general boundary-value problem for the operator lu = −u′′ + q(x)u , 0 < x < 1, If the complex-valued coefficients q(x) is summable on (0,1), the integral $$ {\int}_0^1x\left(1-x\right)\left|q(x)\right| dx $$ converges. The asymptotic solutions of the equation lu = μ2u derived in this article are used to construct the asymptotic spectrum of the problem, to classify the boundary conditions, and to prove theorems asserting that the system of root functions is complete and forms an unconditional basis in L2 (0,1).

Radon transforms on Siegel-type nilpotent Lie groups

Let $$\mathfrak{N}:=H_n\times\mathbb{C}^n$$ be the Siegel-type nilpotent group, which can be identified as the Shilov boundary of Siegel domain of type II, where Hn denotes the set of all n × n Hermitian matrices. In this article, we use singular convolution operators to define Radon transform on $$\mathfrak{N}$$ and obtain the inversion formulas of Radon transforms. Moveover, we show that Radon transform on $$\mathfrak{N}$$ is a unitary operator from Sobolev space Wn,2 into $$L^2(\mathfrak{N})$$ .

Generalized singular value thresholding operator based nonconvex low-rank and sparse decomposition for moving object detection

Moving object detection is one of the most challenging tasks in computer vision and many other fields, which is the basis for high-level processing. Low-rank and sparse decomposition (LRSD) is widely used in moving object detection. The existing methods primarily address the LRSD problem by exploiting the approximation of rank functions and sparse constraints. Conventional methods usually consider the nuclear norm as the approximation of the low-rank matrix. However, the actual results show that the nuclear norm is not the best approximation of the rank function since it simultaneously minimize all the singular values. In this paper, we exploit a novel nonconvex surrogate function to approximate the low-rank matrix and propose a generalized formulation for nonconvex low-rank and sparse decomposition based on the generalized singular value thresholding (GSVT) operator. And then, we solve the proposed nonconvex problem via the alternating direction method of multipliers (ADMM), and also analyze its convergence. Finally, we give numerical results to validate the proposed algorithm on both synthetic data and real-life image data. The results demonstrate that our model has superior performance. And we use the proposed nonconvex model for moving objects detection, and provide the experimental results. The results show that the proposed method is more effective than representative LRSD based moving objects detection algorithms.

Separability properties of singular degenerate abstract differential operators and applications

We study separability and spectral properties of singular degenerate elliptic equations in vector-valued $L_{p}$ spaces. We prove that a realization operator according to this equation with some boundary conditions is separable and Fredholm in $L_{p}$. The leading part of the associated differential operator is not self-adjoint. The sharp estimate of the resolvent, discreteness of spectrum and completeness of root elements of this operator is obtained. Moreover, we show that this operator is positive and generates a holomorphic $C_{0}$-semigroups on $L_{p}$. In application, we examine the regularity properties of nonlocal boundary value problem for degenerate elliptic equation and for the system of degenerate elliptic equations of either finite or infinite number.

Norms of the Positive Powers of the Bessel Operator in the Spaces of Even Schlömilch j-Polynomials

The definition of a B-derivative is based on the notion of generalized Poisson shift; this derivative coincides, up to a constant, with the singular Bessel differential operator. We introduce the fractional powers of a B-derivative by analogy with fractional Marchaud and Weyl derivatives. We prove statements on the coincidence of these derivatives for the classes of even smooth integrable functions. We obtain analogs of Bernstein’s inequality for B-derivatives of integer and fractional order in the space of even Schlomilch j-polynomials with sup-norm and L p γ -norm (the Lebesgue norm with power weight xγ, γ > 0). The resulting estimates are sharp and define the norms of powers of the Bessel operator in the spaces of even Schlomilch j-polynomials.

Biorthogonal Systems of Analytic Functions Generated by a Regular Triangle

We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.