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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]

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]

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]

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]

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 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]

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 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]

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]

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]

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 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]

Let L k = - Δ k + V be the Dunkl–Schrödinger operators, where Δ k = ∑ j = 1 d T j 2 is the Dunkl Laplace operator associated to the Dunkl operators T j on R d and V is a nonnegative potential function. In the first part of this paper we introduce the Riesz transform R j = T j L k - 1 / 2 as a L 2 -bounded operator and we first prove that it is of weak type (1, 1) and then that it is bounded on L p (R d , d μ k (x)) for 1 < p ≤ 2 . The second part is devoted to the L p smoothing of the semigroup generated by L k , when V belongs to the standard Kato class. [ABSTRACT FROM AUTHOR]

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan xθ theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the \sin \theta theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions. [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. [ABSTRACT FROM AUTHOR]

Let Bs (H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dimH ≥ 3. It is proved that a continuous surjective map ϕ on Bs (H) preserves nonzero projections of Jordan products of two operators in both directions if and only if there exist a unitary or an anti-unitary operator U on H and a constant λ with λ² = 1 such that ϕ(A) = λU*AU for all A ∈ Bs (H). [ABSTRACT FROM AUTHOR]

In this work, conditions for the regular solvability of one boundary-value problem for fourth-order operator-differential equations with a variable coefficient on the semi-axis are found. The obtained conditions are expressed by the properties of the coefficients of the considered operator-differential equation. Moreover, the norms of intermediate derivatives operators are estimated in terms of the norm of the right-hand side of the equation and these estimates are related to the solvability conditions of the boundary value problem. [ABSTRACT FROM AUTHOR]

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm–Liouville problems, differential models with involution, fractional Sturm–Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod. [ABSTRACT FROM AUTHOR]

The bi-orthogonality relation for eigenfunctions of self-adjoint operators is derived. Its composition is explained in view of the structure of a characteristic equation and of the energy flow components. Application of the bi-orthogonality relation for solving forcing problems is generalized and the connection between the bi-orthogonality relation and the virtual wave method is highlighted. Technicalities are illustrated in a non-trivial example of propagation of free/forced cylindrical waves in a thin elastic plate under heavy fluid loading. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'. [ABSTRACT FROM AUTHOR]

We discuss basic theoretical principles underlying new numerical methods of calculation of eigenvalues and eigenfunctions of discrete operators semi-bounded from below. We present algorithms of finding spectral characteristics by methods of regularized traces and examples related to certain spectral Sturm–Liouville problems. [ABSTRACT FROM AUTHOR]

In this article, we discuss a conformable fractional Sturm‐Liouville boundary‐value problem. We prove an existence and uniqueness theorem for this equation and formulate a self‐adjoint boundary value problem. We also construct the associated Green function of this problem, and we give the eigenfunction expansions. Finally, we will give some examples. [ABSTRACT FROM AUTHOR]

Given Hilbert space operators such that ( the closure of the numerical range of S), the similarities for invertible A and have been considered by a number of authors over past few decades. A classical result of C. R. De Prima (resp., I. H. Sheth) says that if A and are normaloid or convexoid (resp., A is hyponormal), then implies A is unitary (resp., implies A is self-adjoint). This paper uses (Putnam-Fuglede theorem type) commutativity results to obtain generalizations of extant results on similarities of the above type. Amongst other results, it is proved that if with A invertible and , then: (i) A normaloid implies either A is unitary or ; (ii) operators A satisfying the positivity condition are unitary. If the operator A in (resp., ) is w-hyponormal or class with , then a sufficient condition for A to be unitary (resp., A to be self-adjoint) is that ; furthermore, one may drop the hypothesis in the case in which . [ABSTRACT FROM AUTHOR]

A reproducing kernel Hilbert space is a Hilbert space of complex-valued functions on a (non-empty) set Ω, which has the property that point evaluation is continuous on for all . Then the Riesz representation theorem guarantees that for every there is a unique element such that for all . The function is called the reproducing kernel of and the function is the normalized reproducing kernel in . The Berezin symbol of an operator A on a reproducing kernel Hilbert space is defined by The Berezin number of an operator A on is defined by The so-called Crawford number is defined by We also introduce the number defined by It is clear that By using the Hardy-Hilbert type inequality in reproducing kernel Hilbert space, we prove Berezin number inequalities for the convex functions in Reproducing Kernel Hilbert Spaces. We also prove some new inequalities between these numerical characteristics. Some other related results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument. [ABSTRACT FROM AUTHOR]

In this paper, we obtain suffcient conditions for the well-posed and unique solvability in a weighted Sobolev space for a class of inverse parabolic operator-differential equations of third order. The main part of the equation under consideration has a multiple characteristic. We establish a connection between the solvability conditions and the values of the norms of intermediate derivatives operators. These norms are estimated with respect to the norm of the operator generated by the main part of considered equation. The obtained results show the role of the lower boundary of the spectrum of an abstract operator appearing in the main part of the equation. [ABSTRACT FROM AUTHOR]

We describe the self-adjoint realizations of the operator H:=-iα·∇+mβ+V(x), for m∈R, and V(x)=|x|-1(νI4+μβ-iλα·x/|x|β), for ν,μ,λ∈R. We characterize the self-adjointness in terms of the behavior of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension. [ABSTRACT FROM AUTHOR]

Abstract The space of entire functions which are integrable with respect to the Gaussian weight, known also as the Fock space, is one of the preferred functional Hilbert spaces for modeling and experimenting harmonic analysis, quantum mechanics or spectral analysis phenomena. This space of entire functions carries a three parameter family of canonical isometric involutions. We characterize the linear differential operators acting on Fock space which are complex symmetric with respect to these conjugations. In parallel, as a basis of comparison, we discuss the structure of self-adjoint linear differential operators. The computation of the point spectrum of some of these operators is carried out in detail. [ABSTRACT FROM AUTHOR]

We give short proofs of two descriptions given by Šemrl of order automorphisms of the effect algebra. This sheds new light on both formulas that look quite complicated. Our proofs rely on Molnár's characterization of order automorphisms of the cone of all positive operators. [ABSTRACT FROM AUTHOR]

In this paper, we study some complementary inequalities to Jensen's inequality for self-adjoint operators, unital positive linear mappings, and real valued twice differentiable functions. New improved complementary inequalities are presented by using an improvement of the Mond-Pečarić method. These results are applied to obtain some inequalities with quasi-arithmetic means. [ABSTRACT FROM AUTHOR]

In this remark we study the boundary-value problems for a fractional analogue of the Laplace equation with integral boundary conditions in rectangular and half-strip domains. We prove the existence and uniqueness of solutions by using the spectral decomposition method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a q-analog of 1-dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of [ABSTRACT FROM AUTHOR]

This paper presents 2 new classes of the Bessel functions on a compact domain [0, T] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders. [ABSTRACT FROM AUTHOR]

In this paper, a new definition of majorization for C*-algebras is introduced. Sherman's inequality is extended to self-adjoint operators and positive linear maps by applying the method of premajorization used for com- paring two tuples of objects. A general result in a matrix setting is established. Special cases of the main theorem are studied. In particular, a HLPK-type inequality is derived. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the solution (f,g) of the equation = , x,y ∣ X, where f:X→Y , g:X→X, X,Y are inner product spaces over the same field K ∣ {R,C}. [ABSTRACT FROM AUTHOR]

Letfbe a continuous convex function on an intervalJandbe Hilbert spaces. Letbe self-adjoint operators with spectra in [m, M] for some scalarsandbe self-adjoint operators with spectra in. Ifis ak-tuple andis an-tuple of positive linear mappings fromintowith,and, then we prove the inequalities As applications, we prove Jensen’s inequality for operators without operator convexity and obtain some operator inequalities of Jensen type. [ABSTRACT FROM AUTHOR]

We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator- and Coulomb-like potentials) on pseudosphere. We construct all self-adjoint Schrödinger operators for these theories and present solutions of the corresponding spectral problems. We show, that there is one-to-one correspondence between spectral points of the energy-coupling constant planes not only for discrete, but also for continuous spectra and corresponding eigenfunctions forming the complete orthonormalized systems. [ABSTRACT FROM AUTHOR]

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L 2 ( − 1 , 1 ) which has the Legendre polynomials { P n } n = 0 ∞ as eigenfunctions. As a consequence, they explicitly determined the domain D ( A 2 ) of the self-adjoint operator A 2 . However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point x = ± 1 in L 2 ( − 1 , 1 ) so D ( A 2 ) should exhibit four boundary conditions. In this paper, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman–Krein–Naimark) theory. In addition, we determine a new characterization of D ( A 2 ) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple – and natural – and are equivalent to the boundary conditions obtained from the GKN theory. [ABSTRACT FROM AUTHOR]

In this paper we characterize self-adjoint and normal composition operators on Poisson weighted sequence spaces l²(λ). However, the main purpose of this paper is to determine explicit conditions on inducing map under which a composition operator admits a best normal approximation. We extend results of Tripathi and Lal [Antinormal composition operators on l², Tamkang J. Math. 39 (2008), 347-352] to characterize antinormal composition operators on l²(λ). [ABSTRACT FROM AUTHOR]

In this article we consider the Robin-Cauchy problem for multi-dimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Robin-Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered Robin-Cauchy problem. It is shown that the ill-posedness of the Robin-Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with the deviating argument. [ABSTRACT FROM AUTHOR]

We generalize to several variables the classical theorem of Nevanlinna that characterizes the Cauchy transforms of positive measures on the real line. We show that for the Loewner class, a large class of analytic functions that have non-negative imaginary part on the upper polyhalf-plane, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space. We find four different representation formulae and we show that every function in the Loewner class has one of the four representations, corresponding precisely to four different growth conditions at infinity. [ABSTRACT FROM AUTHOR]

Some sharp bounds for the Čebyšev functional of a function that is convex in absolute value and applications for functions of self-adjoint operators in Hilbert spaces via the spectral representation theorem are given. [ABSTRACT FROM AUTHOR]