Singular Operators Research Articles

Strictly singular non-compact operators between $L_p$ spaces

We study the structure of strictly singular non-compact operators between L_p spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators, it is shown that there exist operators T , for which the set of points (1/p,1/q)\in(0,1)\times (0,1) such that T\colon L_p\rightarrow L_q is strictly singular but not compact contains a line segment in the triangle \{(1/p,1/q):1<p<q<\infty\} of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of L_p is also explored.

Local grand Lebesgue spaces on quasi-metric measure spaces and some applications

We introduce local grand Lebesgue spaces, over a quasi-metric measure space \( ( X,d, \mu ) \), where the Lebesgue space is “aggrandized” not everywhere but only at a given closed set F of measure zero. We show that such spaces coincide for different choices of aggrandizers if their Matuszewska–Orlicz indices are positive. Within the framework of such local grand Lebesgue spaces, we study the maximal operator, singular operators with standard kernel, and potential type operators. Finally, we give an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain.

Analysis of non-singular fractional bioconvection and thermal memory with generalized Mittag-Leffler kernel

This paper deals with the application of non-singular fractional operator in the bioconvection flow of a MHD viscous fluid for vertical surface. The Laplace transform method is used for dimensionless governing equations of momentum, energy and diffusion respectively. Classical governing model is extended to non-integer order approach with non-singular kernel which can be used to describe the memory for natural phenomena. The main advantage is to use this fractional operator can it measure the rate of change at all points of the considered interval, therefore, the present fractional operator incorporate the previous history/memory effects of any system. For the prediction of physical behavior of embedded parameters, some graphs are presented in the graphical section. At the end some remarkable results are found. It is found that non-singular fractional operator measures the memory better in comparison with singular fractional operator. Further, on comparison between different kinds of viscous fluid (Water, Air, Kerosene), it is found that temperature and velocity of air is higher than water and kerosene respectively. The results are validated with the recent published work.

Weighted Boundedness of Certain Sublinear Operators in Generalized Morrey Spaces on Quasi-Metric Measure Spaces Under the Growth Condition

We prove weighted boundedness of Calderón–Zygmund and maximal singular operators in generalized Morrey spaces on quasi-metric measure spaces, in general non-homogeneous, only under the growth condition on the measure, for a certain class of weights. Weights and characteristic of the spaces are independent of each other. Weighted boundedness of the maximal operator is also proved in the case when lower and upper Ahlfors exponents coincide with each other. Our approach is based on two important steps. The first is a certain transference theorem, where without use homogeneity of the space, we provide a condition which insures that every sublinear operator with the size condition, bounded in Lebesgue space, is also bounded in generalized Morrey space. The second is a reduction theorem which reduces weighted boundedness of the considered sublinear operators to that of weighted Hardy operators and non-weighted boundedness of some special operators.

New classes of function spaces and singular operators

This article is dedicated to the memory of Garnik Al’bertovich Karapetyan and it contains a review of results obtained by G. A. Karapetyan and his colleagues within the joint Russian–Armenian project of RFBR. In the first section, we look at multi-anisotropic spaces which were intensively studied by Karapetyan and his students. The second section is devoted to a new class of singular Hausdorff and Hausdorff–Berezin operators. In the third section, we study the connection between real function spaces and operator algebras in a Hilbert space, established by means of a quantization procedure. UDK: 517.518.

THE MAXIMAL IDEAL IN THE SPACE OF OPERATORS ON

AbstractWe study the isomorphic structure of $(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$ and prove that these spaces are complementably homogeneous. We also show that for any operator T from $(\sum {\ell }_{q})_{c_{0}}$ into ${\ell }_{q}$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ that is isometric to $(\sum {\ell }_{q})_{c_{0}}$ and the restriction of T on X has small norm. If T is a bounded linear operator on $(\sum {\ell }_{q})_{c_{0}}$ which is $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular, then for any $\epsilon>0$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ which is isometric to $(\sum {\ell }_{q})_{c_{0}}$ with $\|T|_{X}\|<\epsilon $ . As an application, we show that the set of all $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular operators on $(\sum {\ell }_{q})_{c_{0}}$ forms the unique maximal ideal of $\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$ .

Singular integral operators in the case of a piecewise Lyapunov contour

The article attempts to present the results obtained by the author in recent years (in a number of cases with some additions) on the theory of singular integral equations and Riemann boundary value problems in the case of a piecewise Lyapunov contour. It has a survey character of problems and results related to the influence of corner points of integration contour on various properties of singular operators. At the same time, much attention is paid to the research of other authors related to the scientific interests of the author of this work.

On some approximation properties of Gauss-Weierstrass singular operators

Approximation theorems were formulated for function continuous in the neighborhood of some point x, -∞< x < ∞. Namely, the upper bounds were obtained for the function approximations by their Gauss-Weierstrass singular operators in terms of a majorant function for the first- and second-order continuity moduli of the relevant functions.

Quantum integrability of massive anisotropic SU(N) fermionic models

We consider a general anisotropic massive SU(N) fermionic model, and investigate its quantum integrability. In particular, by regularizing singular operator products, we derive a system of equations resulting in the S-matrix and find some non-trivial solutions. We illustrate our findings on the example of a SU(3) model, and show that the Yang-Baxter equation is satisfied in the massless limit for all coupling constants, while in the massive case the solutions are parameterized in terms of the exceptional solutions to the eight-vertex model.

Singular Green operators in the edge algebra formalism

We give a new description of the class of singular Green operators introduced by Grubb in the study of parameter-dependent boundary value problems, employing the concept of operator-valued symbols introduced by Schulze in the framework of pseudodifferential operators on manifolds with edges.

Resolvent and logarithmic residues of a singular operator pencil in Hilbert spaces

The present paper considers the operator pencil A(λ)=A0+A1λ, where A0,A1≠0 are bounded linear mappings between complex Hilbert spaces and A0 is neither one-to-one nor onto. Assuming that 0 is an isolated singularity of A(λ) and that the image of A0 is closed, certain operators are defined recursively starting from A0 and A1 and they are shown to provide a characterization of the image and null space of the operators in the principal part of the resolvent and of the logarithmic residues of A(λ) at 0. The relations with the classical results based on ascent and descent in [10] are discussed. In the special case of A0 being Fredholm of index 0, the present results characterize the rank of the operators in the principal part of the resolvent, the dimension of the subspaces that define the ascent and descent, the partial multiplicities, and the algebraic multiplicity of A(λ) at 0.

ℓ₂-strictly singular operators on the predual of a hyperfinite von Neumann algebra

We show that every ℓ 2 \ell _2 -strictly singular operator on the predual of any atomless hyperfinite finite von Neumann algebra M \mathcal {M} is Dunford–Pettis, which extends a Rosenthal’s theorem for the case of commutative algebra M = L ∞ ( 0 , 1 ) \mathcal {M}=L_\infty (0,1) . We also apply our result to the study of noncommutative symmetric spaces X = E ( M , τ ) X=E(\mathcal {M},\tau ) for which every ℓ 2 \ell _2 -strictly singular operator from L p ( 0 , 1 ) L_p(0,1) into X X is narrow.

Dilation, model, scattering and spectral problems of second-order matrix difference operator

In the Hilbert space ?2 ?(Z; E) (Z := {0,? 1, ? 2, ...}, dim E = N &lt; ?), the maximal dissipative singular second-order matrix difference operators that the extensions of a minimal symmetric operator with maximal deficiency indices (2N, 2N) (in limit-circle cases at ? ?) are considered. The maximal dissipative operators with general boundary conditions are investigated. For the dissipative operator, a self-adjoint dilation and is its incoming and outgoing spectral representations are constructed. These constructions make it possible to determine the scattering matrix of the dilation. Also a functional model of the dissipative operator is constructed. Then its characteristic function in terms of the scattering matrix of the dilation is set. Finally, a theorem on the completeness of the system of root vectors of the dissipative operator is proved.

The Approximation of Weighted Hölder Functions by Fourier-Jacobi Polynomials to the Singular Sturm-Liouville Operator

In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.

A completeness theorem for dissipative conformable fractional Sturm-Liouville operator in singular case

This paper is concerned with the singular dissipative conformable fractional Sturm-Liouville operators. A completeness theorem for these operators is proved.

Singular degenerate normal operators for first-order

Singular degenerate normal operators for first-order

Computational analysis of COVID-19 model outbreak with singular and nonlocal operator

&lt;abstract&gt;&lt;p&gt;The SARS-CoV-2 virus pandemic remains a pressing issue with its unpredictable nature, and it spreads worldwide through human interaction. Current research focuses on the investigation and analysis of fractional epidemic models that discuss the temporal dynamics of the SARS-CoV-2 virus in the community. In this work, we choose a fractional-order mathematical model to examine the transmissibility in the community of several symptoms of COVID-19 in the sense of the Caputo operator. Sensitivity analysis of $ R_{0} $ and disease-free local stability of the system are checked. Also, with the assistance of fixed point theory, we demonstrate the existence and uniqueness of the system. In addition, numerically we solve the fractional model and presented some simulation results via actual estimation parameters. Graphically we displayed the effects of numerous model parameters and memory indexes. The numerical outcomes show the reliability, validation, and accuracy of the scheme.&lt;/p&gt;&lt;/abstract&gt;

ОПРЕДЕЛЕНИЕ НЕПРЕРЫВНОГО ЗАПАЗДЫВАНИЯ В СПЕКТРАЛЬНОЙ ЗАДАЧЕ ДЛЯ ОПЕРАТОРА ЧЕБЫШЁВА ПЕРВОГО РОДА

A perturbed singular ordinary differential Chebyshev operator of the first kind with continuous delay is considered in this paper. For an arbitrary numerical sequence that does not differ much from eigenvalues sequence of an unperturbed operator, a problem is set to find the perturbation operator containing a continuous delay. A theorem of the existence of such an operator is being proved. An algorithm of finding the delay function in the form of a Fourier series is built and substantiated. The algorithm substantiation is based on the regularized traces theory.

Distribution dependent SDEs for Navier-Stokes type equations

To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.

Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type

In this paper, we consider a linear operator of the Korteweg-de Vries type Lu = ?u ?y + R2(y)?3u ?x3 + R1(y)?u ?x + R0(y)u initially defined on C? 0,?(??), where ?? = {(x, y) : ?? ? x ? ?,?? &lt; y &lt; ?}. C? 0,?(??) is a set of infinitely differentiable compactly supported function with respect to a variable y and satisfying the conditions: u(i) x (??, y) = u(i) x (?, y), i = 0, 1, 2. With respect to the coefficients of the operator L , we assume that these are continuous functions in R(??,+?) and strongly growing functions at infinity. In this paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.