Related papers: Combinatorial identities related to eigenfunction …
Heckman-Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal D_i$ as $\mathcal P_m= \sum_{i=1}^N (x_i \mathcal D_i)^m$. They have been known since…
Given a LHS (Lattice of Hilbert spaces) $V_J$ and a symmetric operator $A$ in $V_J$, in the sense of partial inner product spaces, we define a generalized resolvent for $A$ and study the corresponding spectral properties. In particular, we…
We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass…
Motivated by Liu's recent work in \cite{Liu2022}. We shall reveal the essential feature of Hahn polynomials by presenting two new $q$-exponential operators. These lead us to use a systematic method to study identities involving Hahn…
In this paper we derive some identities and inequalities on the M\"obius mu function. Our main tool is phi functions for intervals of positive integers and their unions.
Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can…
In this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the Hilbert scheme of points in the plane.
For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint…
We consider here a new operator, called ``super nabla'', which is shown to be generic among operators for which the modified Macdonald polynomials are joint eigenfunctions. All previously known Macdonald eigenoperators can readily be…
For W*-algebras A and self-dual Hilbert A-modules M we show that every self-adjoint, ''compact'' module operator on M is diagonalizable. Some specific properties of the eigenvalues and of the eigenvectors are described.
This paper investigates the eigenvalue problem of integral operators whose kernels can be expressed as a finite sum of pairwise products of single-variable functions, making them separable. By consdiering the matrix form of the separable…
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…
This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including…
In this paper, we generalize the notion of joint eigenvalues and joint spectrum of matrices and operator tupples on a bi complex Hilbert space. We observe that unlike the spectrum of a bounded operator on a bi complex Hilbert space is…
By using a real matrix translation, we propose a coupled eigenvalue problem for octonionic operators. In view of possible applications in quantum mechanics, we also discuss the hermiticity of such operators. Previous difficulties in…
We show that total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ with kernel $h(t)$ and of an operator of multiplication by some real function $s(x)$ coincide. In particular, $\pm H\geq 0$ if and only…
A certain analysis of all possible associative binary operations on N is presented. This is equivalent with an analysis of all possible monoid structures on N. Several results and a conjecture in this regard are given.
In this paper, we investigate the spectrum of the self adjoint differential operator with operator coefficitent in a separable Hilbert space. We also derive asymptotic formulas for the sum of eigenvalues of this operator.
In this contribution we analyze the spectral properties of some commonly used boundary integral operators in computational electromagnetics and of their discrete counterparts, highlighting peculiar features of their spectra. In particular,…
In a right quaternionic Hilbert space, following the complex formalism, decomposable operators, the so-called Bishop's property and the single valued extension property are defined and the connections between them are studied to certain…