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The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k ≥ 1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates. [ABSTRACT FROM AUTHOR]

We extend the (outer) measure γ I associated to an operator ideal I to a measure γ J for bounded bilinear operators. If I is surjective and closed, and J is the class of those bilinear operators such that γ J (T) = 0 , we prove that J coincides with the composition bideal I ∘ B . If I satisfies the Σr-condition, we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to J . Furthermore, if in addition I is symmetric, we prove a formula for the measure γ J of an operator interpolated by the real method. In particular, results apply to weakly compact operators. [ABSTRACT FROM AUTHOR]

Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderón–Zygmund classes. For any b → = (b 1 , b 2) ∈ (CMO (R n)) 2 , let [ T , b j ] e j (j = 1 , 2) , [ T , b → ] α be the commutators in the j-th entry and the iterated commutators of T, respectively. In this paper, for all p0 > 1, p 0 2 < p < ∞ , and p0 ≤ p1, p2 < ∞ with 1/p = 1/p1 + 1/p2, we prove that [ T , b j ] e j and [ T , b → ] α are weighted compact operators from L p 1 (w 1) × L p 2 (w 2) to L p (ν w → ) , where w → = (w 1 , w 2) ∈ A p → / p 0 and ν w → = w 1 p / p 1 w 2 p / p 2 . As applications, we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity, including bilinear g function, bilinear g*λ function and bilinear Lusin's area integral. In addition, we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear g function, bilinear g*λ function and bilinear Lusin's area integral, respectively. [ABSTRACT FROM AUTHOR]

In this study, we introduce a kind of Stancu variant of the complex Bernstein-Durrmeyer-type operators in movable compact disks. Their approximation properties for analytic functions in the movable compact disks are investigated. [ABSTRACT FROM AUTHOR]

It has become a common challenge on how to measure radiation patterns of massive antenna arrays working at high frequency with high efficiency and accuracy, which has attracted huge interest from both academia and industry. In this paper, a highly accurate and efficient far field (FF) pattern reconstruction strategy through conducting single cut near field (NF) to FF transformation algorithm based on spherical wave expansion (SWE) is proposed for fast measurements of large base stations (BSs) operating at mid-band. Our objective is to design a compact multi-probe setup to measure key radiation parameters of the massive array, with low-cost and high-efficiency. A BS of 16 × 16 elements operating at mid-band is adopted as the device under test (DUT) in this paper. Analysis is conducted on the impact of measurement distance, sampling range and sampling density on the reconstruction accuracy of the algorithm in the compact multi-probe anechoic chamber setup, demonstrating that the required key information of the main lobe of the DUT can be accurately reconstructed when the measurement range R = 1/20 FF distance, sampling resolution ∆θ = 3°, and sampling range θ ∈ [75°, 105°] are selected, respectively. The impact of phase center offset on the accuracy of FF pattern reconstruction is analyzed when only one sub-array is enabled, demonstrating the effectiveness of the algorithm for sub-array measurements. [ABSTRACT FROM AUTHOR]

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms. [ABSTRACT FROM AUTHOR]

For any square-summable commuting family (A i) i ∈ I of complex n × n matrices there is a normal commuting family (B i) i no farther from it, in squared normalized ℓ 2 distance, than the diameter of the numerical range of ∑ i A i ⁎ A i. Specializing in one direction (limiting case of the inequality for finite I) this recovers a result of M. Fraas: if ∑ i = 1 ℓ A i ⁎ A i is a multiple of the identity for commuting A i ∈ M n (C) then the A i are normal; specializing in another (singleton I) retrieves the well-known fact that close-to-isometric matrices are close to isometries. [ABSTRACT FROM AUTHOR]

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v ∇ v in interval (t 0 , t 1) , and linear interpolation concerning two previous time points is employed to deal with v ∇ v in intervals (t n − 1 , t n) , n ≥ 2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method. [ABSTRACT FROM AUTHOR]

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the φ − Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra R and commutative Banach algebra of the collection of continuous functions in R. [ABSTRACT FROM AUTHOR]

First, we give conditions on a Banach lattice E$E$ so that an operator T$T$ from E$E$ to any Banach space is disjoint p$p$‐convergent if and only if T$T$ is almost Dunford–Pettis. Then, we study when adjoints of positive operators between Banach lattices are disjoint p$p$‐convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices E$E$ and F$F$: (i) a positive operator T:E→F$T: E \rightarrow F$ is almost weak p$p$‐convergent if and only if T∗$T^*$ is disjoint p$p$‐convergent; (ii) E∗$E^*$ has order continuous norm or F∗$F^*$ has the positive Schur property of order p$p$. Very recent results are improved, examples are given and applications of the main results are provided. [ABSTRACT FROM AUTHOR]

We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric. [ABSTRACT FROM AUTHOR]

A bounded linear operator T on a Hilbert space H is said to be absolutely norm attaining (T ∈ AN (H)) if the restriction of T to any non-zero closed subspace attains its norm and absolutely minimum attaining (T ∈ AM (H)) if every restriction to a non-zero closed subspace attains its minimum modulus. In this article, we characterize normal operators in AN (H) ¯ , the operator norm closure of AN (H) , in terms of the essential spectrum. Later, we study representations of quasinormal and hyponormal operators in AN (H) ¯ . Explicitly, we prove that any hyponormal operator in AN (H) ¯ is a direct sum of a normal AN -operator and a 2 × 2 upper triangular AM -operator matrix. Finally, we deduce some sufficient conditions implying the normality of them. [ABSTRACT FROM AUTHOR]

Some classes of operators T on are characterized in terms of the induced operators Tn on each Xn. We study pseudo weakly compact, limited completely continuous, q-convergent, DP q-convergent, limited q-convergent, weak Dunford-Pettis, weak* Dunford-Pettis, weak q-convergent, weak* q-convergent operators on for 1 < q < ∞. We also study q-convergent operators (1 ≤ q < p*) and operators with weakly precompact adjoints on. Applications to some properties on the space are given. We study L-limited sets, q-L-limited sets, Right sets, q-Right sets, Right∗ sets, q-Right* sets, L-limited* sets, q-L-limited*, and L-sets in vector sequence spaces. [ABSTRACT FROM AUTHOR]

Let E be a sublattice of a vector lattice F. A continuous operator T from E into a normed vector space X is said to be õrder-norm continuous if ... implies ... for every (xα)α∈A ⊆ E. This paper aims to investigate the properties of this new class of operators and explore their relationships with existing classifications of operators. We introduce a new class of operators called õrder weakly compact operators. A continuous operator T : E → X is considered õrder weakly compact if T(A) in X is a relatively weakly compact set for every Fo-bounded A ⊆ E. In this manuscript, we examine various properties of this class of operators and explore their connections with õrder-norm continuous operators. [ABSTRACT FROM AUTHOR]

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive an explicit graph-theoretic formula for the numerics in heat coefficients as a linear combination of metric jets based on Polterovich’s formula. [ABSTRACT FROM AUTHOR]

A scattered operator is a bounded linear operator with at most countable spectrum. In this paper, we prove that for any elementary operator on B (H) , not only for finite length but also for infinite length, if the range of the elementary operator is contained in scattered operators, then the corresponding sum of multipliers is a compact operator. We also prove that for some special classes of elementary operators, such as the elementary operators of length 2, higher order inner derivations and generalized inner derivation, if the range of the elementary operator is contained in the set of scattered operators, then the range is contained in the set of power compact operators. At the same time, the multipliers of the corresponding elementary operators are characterized. [ABSTRACT FROM AUTHOR]

Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and potentials. In this review, we present some generalisations of substitutions, with a focus on substitutions on compact alphabets, and with an outlook towards their spectral theory. Guided by two main examples, we will illustrate what changes when one moves from finite to compact (infinite) alphabets, and discuss under which assumptions do we recover the usual geometric and statistical properties which make them viable models of materials with almost periodic order. We also present a planar example (which is a two‐dimensional generalisation of the Thue−Morse substitution), whose diffraction is purely singular continuous. [ABSTRACT FROM AUTHOR]

There are two aims in this paper. The first aim is to characterize the order-bounded weighted composition operators between Fock spaces, and the second is to further characterize the order-bounded difference in weighted composition operators between Fock spaces. At the same time, six examples are given to illustrate the relations between boundedness and ordered boundedness. Moreover, an interesting result is found that differences in weighted composition operators defined by some special weighted functions and symbol functions are order-bounded between Fock spaces if and only if each weighted composition operator is compact between Fock spaces. Finally, two open questions are also put forward for converting larger Fock spaces into smaller ones. [ABSTRACT FROM AUTHOR]

Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the |$k$| -th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär's theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a new class of operators related to norm bounded sets on Banach Lattices and which brings together several classes of operators (as o-weakly compact, b-weakly compact, M-weakly compact, L-weakly compact and almost Dunford–Pettis operators). As applications, we find some new lattice approximation properties of these classes of operators. [ABSTRACT FROM AUTHOR]

In this research, we have elaborated high-rate multistep collocation strategies in order to concern with second-type vanishing delay VIEs. Herein, characteristics of uniqueness, existence, and regularity for both numerical and analytical solutions have been shown. To explore the solvability of the system derived from the numerical method, we have defined particular operators and demonstrated that these operators are both compact and bounded. Solvability is studied by means of the innovative compact operator concepts. The concept of convergence has been examined in greater detail, revealing that the convergence of the method is influenced by the spectral radius of the matrix generated according to the collocation parameters in the difference equation resulting from the method's error. Finally, two numerical examples are given to certify our theoretically gained results. Also, since the proposed numerical method is local in nature, it can be compared to other local methods, such as those used in reference [1]. We will compare our method with [1] in the last section. [ABSTRACT FROM AUTHOR]

A family of quasi-1D Schrödinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibits changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C∗-algebra. The quotient of this algebra by the ideal of compact operators is studied, and an index theorem is deduced from these investigations. This result corresponds to a topological version of Levinson’s theorem in the presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the last two sections of the paper, the K-theory of the main C∗-algebra and the dependence on an external parameter are carefully analyzed. In particular, a surface of resonances is exhibited, probably for the first time. The contents of these two sections are of independent interest, and the main result does not depend on them. [ABSTRACT FROM AUTHOR]

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact. [ABSTRACT FROM AUTHOR]

We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one. [ABSTRACT FROM AUTHOR]

Let H be a separable complex Hilbert space, B (H) the algebra of bounded linear operators on H , μ a finite Borel measure on R + with the finite (n + 1) -th moment, f (z) : = ∫ R + e − t z d μ (t) for all ℜ z ⩾ 0 , C Ψ (H) , and | | · | | Ψ the ideal of compact operators and the norm associated to a symmetrically norming function Ψ , respectively. If A , D ∈ B (H) are accretive, then the Laplace transformer on B (H) , X ↦ ∫ R + e − t A X e − t D d μ (t) is well defined for any X ∈ B (H) as is the newly introduced Taylor remainder transformer R n (f ; D , A) X : = f (A) X − ∑ k = 0 n 1 k ! ∑ i = 0 k (− 1) i k i A k − i X D i f (k) (D). If A , D * are also (n + 1) -accretive, ∑ k = 0 n + 1 (− 1) k n + 1 k A n + 1 − k X D k ∈ C Ψ (H) and | | · | | Ψ is Q* norm, then | | · | | Ψ norm estimates for ∑ k = 0 n + 1 n + 1 k A k A n + 1 − k 1 2 R n (f ; D , A) X ∑ k = 0 n + 1 n + 1 k D n + 1 − k D * k 1 2 are obtained as the spacial cases of the presented estimates for (also newly introduced) Taylor remainder transformers related to a pair of Laplace transformers, defined by a subclass of accretive operators. [ABSTRACT FROM AUTHOR]

In this article, the regular Schröder–Catalan matrix is constructed and acquired by benefiting Schröder and Catalan numbers. After that, two sequence spaces are introduced, described as the domain of Schröder–Catalan matrix. Additionally, some algebraic and topological properties of the spaces in question, such as completeness, inclusion relations, basis and duals, are examined. In the last two sections, the necessary and sufficient conditions of some matrix classes and compact operators related aforementioned spaces are presented. [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded, compact, and order bounded sums of weighted composition-differentiation operators from Bergman-type spaces to weighted Banach spaces of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on Lp space for 1 ≤ p ≤∞, are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied. [ABSTRACT FROM AUTHOR]

Recently, some new sequence spaces ℓ p (A α) (0 < p < ∞) , c 0 (A α) , c (A α) , and ℓ ∞ (A α) have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of A α = (a n , v α) , where a m , v α = { v α ρ (α) (m) , v ∣ m , 0 , v ∤ m , and ρ (α) (m) : = sum of the α th power of the positive divisors of m ∈ N . They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes. This article deals with some geometric properties of these sequence spaces. [ABSTRACT FROM AUTHOR]

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained. [ABSTRACT FROM AUTHOR]

The fractional differential equation L^\beta u = f posed on a compact metric graph is considered, where \beta >0 and L = \kappa ^2 - \nabla (a\nabla) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients \kappa,a. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L^{-\beta }. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L_2(\Gamma \times \Gamma)-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for {L = \kappa ^2 - \Delta, \kappa >0} are performed to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze the alternating direction implicit (ADI) difference schemes in conjunction with the second order backward differentiation formula (BDF2) method with variable time step sizes for solving the two-dimensional parabolic equation. The spatial compact operators are also applied to construct high order ADI scheme. By using the discrete energy method and the positive definiteness of the nonuniform BDF2 approximation, we prove the unconditional H 1 semi-norm stability for the variable-step BDF2-ADI scheme and the variable-step compact BDF2-ADI scheme under the constraint r n ≤ 4.8 , where r n denotes the adjacent step size ratio. Moreover, the optimal second order and the fourth order convergence rates are derived rigorously under this restriction. To the best of our knowledge, this is the first strict theoretical analysis of the variable-step ADI numerical schemes for the multidimensional parabolic equations. Several numerical examples are included to verify the analysis results. The extension to the nonlinear Allen-Cahn equation is also presented, for which the variable-step BDF2-ADI scheme and the corresponding compact scheme combined with the adaptive time-stepping algorithm improve the efficiency of the long time simulation. [ABSTRACT FROM AUTHOR]

Many studies have been conducted on statistical convergence, which remains an area of active research. Since its introduction, statistical convergence has found applications in many fields. Nevertheless, there is a shortage of studies related to operator theory; especially of those on continuous, bounded, and compact operators. We explore the notions of statistical boundedness, continuity, and compactness of operators between normed spaces, establishing connections between these concepts and their counterparts in the traditional normed space theory. Additionally, we provide some examples and results that demonstrate the behavior and implications of statistical convergence in operator theory. [ABSTRACT FROM AUTHOR]

We prove a compact version of the T1 theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator T admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal CMO condition, and the product CMO condition, then T can be extended to a compact operator on L^p(w) for all 1

We consider the stabilization of a class of linear evolution systems z ′ = A z + B v under the observation y = C z by means of a finite dimensional control v. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator A is supposed to generate an analytical semigroup with compact resolvent and the operators B and C are unbounded operators whereas in the finite dimensional case, A is assumed to be a self-adjoint operator with compact resolvent, B and C are supposed to be bounded operators. In both cases, we show that if (A, B) and (A, C) verify the Fattorini-Hautus Criterion, then we can construct an observer-based control v of finite dimension (greater or equal than largest geometric multiplicity of the unstable eigenvalues of A) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the diffusion system. [ABSTRACT FROM AUTHOR]

The Black–Scholes (B–S) equation has been recently extended as a kind of tempered time-fractional B–S equations, which becomes an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B–S model. To achieve high-order accuracy in space and overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This study aims to construct the BK-spaces ℓp(M) and ℓ∞(M) as the domains of the conservative Motzkin matrix M obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses α-, β-, and γ-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness. [ABSTRACT FROM AUTHOR]

This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n×nn\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

This study aims to construct the BK-spaces $ \ell_p(\mathcal{M}) $ and $ \ell_{\infty}(\mathcal{M}) $ as the domains of the conservative Motzkin matrix $ \mathcal{M} $ obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses $ \alpha $-, $ \beta $-, and $ \gamma $-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.

The article discusses the existence of weak solutions of the Kelvin–Voigt model in the context of the motion of water with added polymers. The model considers the delay in establishing equilibrium states due to relaxation time and internal reorganization processes. The study aims to prove the weak solvability of the initial–boundary value problem by introducing regular Lagrangian flows and defining weak solutions based on a priori estimates and the theory of the topological degree of contracting vector fields. The research was funded by the Russian Science Foundation and the author declares no conflicts of interest. [Extracted from the article]

We investigate vacuum expectation value of the energy-momentum tensor for a massive Dirac field in flat spacetime with a toroidal subspace of a general dimension. Quasiperiodicity conditions with arbitrary phases are imposed on the field operator along compact dimensions. These phases are interpreted in terms of magnetic fluxes enclosed by compact dimensions. The equation of state in the uncompact subspace is of the cosmological constant type. It is shown that, in addition to the diagonal components, the vacuum energy-momentum tensor has nonzero off-diagonal components. In special cases of twisted (antiperiodic) and untwisted (periodic) fields the off diagonal components vanish. For untwisted fields the vacuum energy density is positive and the energy-momentum tensor obeys the strong energy condition. For general values of the phases in the periodicity conditions the energy density and stresses can be either positive or negative. The numerical results are given for a Kaluza-Klein type model with two extra dimensions. [ABSTRACT FROM AUTHOR]

Let \Omega be a bounded pseudoconvex domain in \mathbb {C}^n with Lipschitz boundary and \phi be a continuous function on \overline {\Omega }. We show that the Toeplitz operator T_{\phi } with symbol \phi is compact on the weighted Bergman space if and only if \phi vanishes on the boundary of \Omega. We also show that compactness of the Toeplitz operator T^{p,q}_{\phi } on \overline {\partial }-closed (p,q)-forms for 0\leq p\leq n and q\geq 1 is equivalent to \phi =0 on \Omega. [ABSTRACT FROM AUTHOR]

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace's equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in R d , d ≥ 2 , in the space L 2 (Γ) , where Γ denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace's equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ) . Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in L 2 (Γ) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations. [ABSTRACT FROM AUTHOR]

We establish conditions for the existence of solutions of discrete equations with precompact range of values by using c-continuous operators and admissible pairs of compact sets. [ABSTRACT FROM AUTHOR]

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results. [ABSTRACT FROM AUTHOR]

In this paper, the authors completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Their main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras. [ABSTRACT FROM AUTHOR]

This paper studies the basic topological properties of weighted composition operators on the weighted sequence spaces , with , given by a weight sequence of positive reals such as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms, and description of closed range operators. Previously these properties were studied by Luan and Khoi in the case of Hilbert space . Their methods can be also applied with some minor modifications to the case of Banach spaces with . They based essentially on using the dual spaces of continuous linear functionals and, consequently, cannot be applied to the quasi-Banach case . Moreover, some of them do not work even in . Motivated by these reasons, we develop a more universal approach that allows studying the whole scale of spaces . To this end, we establish the necessary and sufficient conditions for a linear operator to be compact on an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces. Moreover, we introduce the new characteristic, the -essential norm of a continuous linear operator on a quasi-Banach space . This characteristic measures the distance in the operator metric, between and the set of all -compact operators on . Here an operator is -compact on if is compact and coordinatewise continuous on . We show that for with the essential and -essential norms of a weighted composition operator coincide, whereas for we do not know whether the same is true or not. Our main results for weighted composition operators in are as follows: We provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operators with compact difference; as well as some exact formula for the -essential norm. Some key aspects of our approach can be used for other operators and scales of spaces. [ABSTRACT FROM AUTHOR]

In this paper we present necessary and sufficient conditions characterizing compact sets in the spaces of Nuclear and Hilbert-Schmidt operators on Hilbert spaces. Based on these results, we also characterize compact sets in the space of probability measures on separable Hilbert spaces. These results are then used in the study of optimal feedback control theory for stochastic systems. We prove existence of optimal feedback control laws for standard and several nonstandard control problems. Further, we present necessary conditions of optimality and an algorithm and related convergence theorem whereby optimal control laws can be constructed. We present also a result characterizing compact sets in the space of probability measures on separable Hilbert spaces. These results have interesting applications in control theory on infinite dimensional Banach spaces as presented in the last section. [ABSTRACT FROM AUTHOR]

In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without any algebraic structure required. Furthermore, this interpolation method for metric spaces when applied to normed spaces is equivalent to the K-method, which has been widely studied in the literature. As an application, we interpolate Fréchet sequence spaces. [ABSTRACT FROM AUTHOR]