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Quasidiagonal operators on a Hilbert space are a large and important class (containing all self-adjoint operators for instance). They are also perfectly suited for study via the finite section method (a particular Galerkin method). Indeed,…
Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical…
Many interesting families of polynomials are indexed by permutations or related objects, and are defined by applying divided difference operators, modified by polynomials, on some initial base case. The fact that these constructions produce…
We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems respectively with a third and a fourth order ladder operators satisfying…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…
First-order automatic differentiation is a ubiquitous tool across statistics, machine learning, and computer science. Higher-order implementations of automatic differentiation, however, have yet to realize the same utility. In this paper I…
We provide a survey of the current state of the study of diagonals of operators, especially selfadjoint operators. In addition, we provide a few new results made possible by recent work of M\"uller-Tomilov and Kaftal-Loreaux. This is an…
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…
This paper addresses a construction of new $q-$Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order…
Self-adjoint rank two commuting ordinary differential operators are studied in this paper. Such operators with trigonometric, elliptic and rapid decay coefficients corresponding to hyperelliptic spectral curves are constructed. Some…
We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations…
Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance, and is associated with operators involving nonlinear combinations of creation and annihilation operators. Here we…
We collect and organise known results and add some new ones of the following nature: if A is a bounded operator in a Hilbert or Banach space, does there exist a nonconstant polynomial p(z) such that p(A) is "simpler", "nicer" than A. The…
We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately.…
We initiate the study of a natural generalisation of the classical Bochner-Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the…
We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are…
Given a sequence of polynomials $(p_n)_n$, an algebra of operators $\mathcal A$ acting in the linear space of polynomials and an operator $D_p\in \mathcal A$ with $D_p(p_n)=\theta_np_n$, where $\theta_n$ is any arbitrary eigenvalue, we…