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Many researchers are interested in topics related to solutions of differential equations with variable coefficients. Here for the first time, we study the nonhomogeneous second-order Euler–Cauchy operator-differential equation. In this paper, we apply the Laplace transform method to find the general solution of the Euler–Cauchy equation with Neumann boundary conditions. We show that this method is powerful in finding the general solution of the Euler–Cauchy equation in terms of generalized exponential functions.

We introduce a new norm on C 2 × C 2 , where C 2 is the Hilbert-Schmidt class. We study basic properties of this norm and prove inequalities involving it. As an application of the present study, we deduce a chain of new bounds for the Hilbert-Schmidt numerical radii of 2 × 2 operator matrices. Connection with the classical Hilbert-Schmidt numerical radius of a single operator are also provided. Moreover, we refine some related existing bounds, too. [ABSTRACT FROM AUTHOR]

The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented. [ABSTRACT FROM AUTHOR]

In this paper, we mainly show that if a product AB (or BA) of a closed symmetric operator A and a bounded positive operator B is normal, then it is self-adjoint. Equivalently, this means that B commutes with A. Certain generalizations and consequences are also presented. [ABSTRACT FROM AUTHOR]

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains Ω ⊂ ℝ d with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by H 0 s (Ω ¯) , and Sobolev–Slobodeckij̆ spaces, denoted by H s (Ω) , where s > d 2 . Our goal is accomplished by obtaining the Green's solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green's kernels satisfying these problems are symmetric and positive definite reproducing kernels of H 0 s (Ω ¯) and H s (Ω) , respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains. [ABSTRACT FROM AUTHOR]

The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R × R + in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above. [ABSTRACT FROM AUTHOR]

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator. In the non-critical case, we do so by providing a boundary triple, and in the critical purely magnetic case, by exploiting the phenomenon of confinement and super-symmetry. Moreover, we justify our model by showing that Dirac operators with singular interactions are limits in the strong resolvent sense of Dirac operators with regular potentials. [ABSTRACT FROM AUTHOR]

We study the convergence of a family of numerical integration methods where the numerical integration is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for a wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function. [ABSTRACT FROM AUTHOR]

In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: ω p T 1 , ⋯ , T n : = sup x = 1 ∑ i = 1 n T i x , x p 1 / p , p ≥ 1 , for all Hilbert space operators T 1 , ⋯ , T n . Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta's inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of n = 1 , the resulting inequalities could be considered extensions and generalizations of the classical numerical radius. [ABSTRACT FROM AUTHOR]

In this work, we study the spaces of generalized elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalization of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements.

We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.

n this paper, aq-analogue of the SturmLiouville problem with discontinuity condition on a finite interval is studied. It is proved that theq-SturmLiouville problem with discontinuity conditions is self-adjoint inLq2(0,). The completeness theorem and the sampling theorem are proved.

Hankel operators have numerous realizations and form one of the most important classes of operators in the spaces of analytic functions. These operators can be defined as those having Hankel matrices (i.e. matrices whose entries depend only on the sum of the indices) with respect to some orthonormal basis for a separable Hilbert space. This paper continues the authors' research of 2021 which introduced a new class of operators in Hilbert spaces; i.e., -Hankel operators, with a complex parameter. These operators act in a separable Hilbert space and have matrices in some orthonormal basis of the space whose diagonals orthogonal to the main diagonal present geometric progressions with common ratio . Thus, the classical Hankel operators correspond to the case . The main result of the article is the normality criterion for -Hankel operators. By analogy with Hankel operators, the class of operators under consideration has concrete realizations in the form of integral operators which enables us to apply the abstract results, and thereby contribute to the theory of integral operators. We consider an example of these realizations in the Hardy space on the unit disk. Also, we give some criteria for the self-adjointness and normality of these operators. [ABSTRACT FROM AUTHOR]

In this work, we give refinements of some well-known numerical radius inequalities. Also, we present an improvement of the triangle inequality for the operator norm. [ABSTRACT FROM AUTHOR]

In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.

Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation Tx=N(x)Tx=N\left(x), where TT is a self-adjoint operator in a real Hilbert space ℋ{\mathcal{ {\mathcal H} }} and NN is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

In this paper, we study the basic theory of regular Hahn-Hamiltonian systems. In this context, we establish an existence and uniqueness result. We introduce the corresponding maximal and minimal operators for this system and some properties of these operators are investigated. Moreover, we give a criterion under which these operators are self-adjoint. Finally, an expansion theorem is proved. [ABSTRACT FROM AUTHOR]

The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R×R+ in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above.

The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfies a system of differential equations with partial derivatives. It is known that a kernel given in a subdomain of the real plane, generally speaking, does not always imply an extension to the entire plane. This possibility is related to the problem of the existence of a commuting selfadjoint extension of symmetric operators. The author applies his own results related to a commuting self-adjoint extension in a wider Hilbert space. The resulting representation in the form of an integral of elementary positive-definite kernels with respect to the spectral measure generated by the resolution of the identity of the operators allows us to extend the positive-definite kernel to the entire plane. [ABSTRACT FROM AUTHOR]

It is shown that every self-adjoint operator in a σ -finite von Neumann factor of type III can be written as a real linear combination of 3 projections. This is in stark contrast to factors of type I ∞ , I I 1 and I I ∞ , where there are always operators that require at least 4 projections in such a representation. [ABSTRACT FROM AUTHOR]

In this article, we investigate some basic results of self-adjoint operator in Γ-Hilbert space. We proof some similar results on self-adjoint operator in this space with some specific norm. Finally we will prove that the spectral theorem for compact self-adjoint operator in Γ-Hilbert space and the converse is also true.

In this paper, a -Dirichlet–Neumann problem with discontinuity conditions on a finite interval is studied. It is proved that the -Dirichlet–Neumann problem with discontinuity conditions is self-adjoint in . Finally, it is shown that eigenfunctions of this problem are in the form of a complete system. [ABSTRACT FROM AUTHOR]

The paper considers a variable third-order operator-differential equation in a separable Hilbert space. Under certain assumptions, it is proved that this ODE has a unique solution. The proof is based on a classical Galerkin discretization of the separable Hilbert space in term of certain eigenfunctions. The approximation quality of the Galerkin approximations can be controlled in terms of the eigenvalues. We deduce estimates for the convergence rate of the approximate solutions to the exact one. An example provided as application to the investigated method. [ABSTRACT FROM AUTHOR]

In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: ωpT1,⋯,Tn:=supx=1∑i=1nTix,xp1/p,p≥1, for all Hilbert space operators T1,⋯,Tn. Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta’s inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of n=1, the resulting inequalities could be considered extensions and generalizations of the classical numerical radius.

In this paper, we consider an impulsive Sturm-Lioville problem on Sturmian time scales. We investigate the existence and uniqueness of the solution of this problem. We study some spectral properties and self-adjointness of the boundary-value problem. Later, we construct the Green function for this problem. Finally, an eigenfunction expansion is obtained. [ABSTRACT FROM AUTHOR]

We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight e−x², and for the best approximation of functions of self-adjoint operators in Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this work, we study the spaces of generalized elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalization of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements. [ABSTRACT FROM AUTHOR]

This paper is focused on Jensen's inequality and its variants. Various refinements and reverses of Jensen's inequality, including scalar and operator versions, are given. [ABSTRACT FROM AUTHOR]

This Note describes various analytical and computational results concerning the calculation of Dirac eigenvalues, or more generally, of operators with gaps. An algorithm based on an abstract theorem characterizing the eigenvalues in gaps was found years ago, but it is only recently that a delicate analysis to identify and study the domains of those operators has allowed to put that algorithm on a firm basis concerning the choice of approximation basis sets, and this both for light and for heavy atoms. The works described here concern joint papers with several collaborators: J. Dolbeault, M. Lewin, M. Loss, E. Séré and M. Vanbreugel.

In this paper, we consider the existence and uniqueness of the generalized solution of a second order partial operator-differential equation in Hilbert space. To this end, at first we prove lemmas on possibility of continuation of some functional generated by the minor terms of the equation on all the Hilbert space. Then we prove a theorem on the existence of a unique generalized solution of a binomial non-homogeneous equation. Using this theorem we prove the existence and uniqueness of the generalized solution of the given operator-differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green's function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L² ?,q((?0, a); E). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to discuss operator Jensen inequality for convex functions, without appealing to operator convexity. Several variants of this inequality will be presented, and some applications will be shown too.

In this work, some generalized Euclidean operator radius inequalities are established. Refinements of some well-known results are provided. Among others, some bounds in terms of the Cartesian decomposition of a given Hilbert space operator are proven.

Abstract In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

In this paper, we establish some Jensen-type inequalities for continuous functions of self-adjoint operators on complex Hilbert spaces. Furthermore, using the Cartesian decomposition of an operator, we improve the known result due to Mond and Peˇcari´c. Some refinements of the H¨older-McCarthy inequality are given as well.

This paper presents two new classes of Müntz functions which are called Jacobi-Müntz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections and their error bounds are derived. Also, two new Müntz type quadrature rules are introduced. As two applications of these basis functions some fractional ordinary and partial differential equations are considered and numerical results are given. [ABSTRACT FROM AUTHOR]

In this work, we study certain spectral properties of the one dimensional Dirac systems on time scales, such as formally self-adjointness, orthogonality of eigenfunctions, Green's function, the existence of a countable sequence of eigenvalues. Later, we give an expansion formula in eigenfunctions for Dirac operators on time scales. These results could provide an important contribution to the spectral theory of such operators on time scales. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a second regularized trace formula on L2([0, π]; H) for a higher order self‐adjoint differential operator with unbounded operator‐valued coefficient, where H is a separable Hilbert space. [ABSTRACT FROM AUTHOR]

We investigate spectral properties of ordinary differential operators related to expressions of the form D∈ + a. Here a ∈ R and D∈ denotes a composition of δ and δ+ according to the signs in the multi-index ∈, where is a first order linear differential expression, called delta-derivative, and δ+ is its formal adjoint in an appropriate L2 space. In particular, Sturm-Liouville operators that admit the decomposition of the type δ+δ + a are considered. We propose an approach, based on weak delta-derivatives and delta-Sobolev spaces, which is particularly useful in the study of the operators D&#8712: + a. Finally we examine a number of examples of operators, which are of the relevant form, naturally arising in analysis of classical orthogonal expansions. [ABSTRACT FROM AUTHOR]