Related papers: Some binomial formulas for non-commuting operators
A d-contraction is a d-tuple $(T_1,...,T_d)$ of mutually commuting operators acting on a common Hilbert space H such that $ \|T_1\xi_1+T_2\xi_2+... +T_d\xi_d\|^2\leq \|\xi_1\|^2+\|\xi_2\|^2+...+\|\xi_d\|^2 $ for all…
In a recent paper a pair of operators $a$ and $b$ satisfying the equations $a^\dagger a=bb^\dagger+\gamma\1$ and $aa^\dagger=b^\dagger b+\delta\1$, has been considered, and their nature of ladder operators has been deduced and analysed.…
Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized…
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are…
We consider differential operators over a noncommutative algebra $A$ generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on $A$-modules $E$ with covariant derivative. We use…
I revisit the so called "bispectral problem" introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues,…
A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the…
We provide the solution to the normal ordering problem for powers and exponentials of two classes of operators. The first one consists of boson strings and more generally homogeneous polynomials, while the second one treats operators linear…
Quantum calculus based on the right invertible divided difference operator $D_{\sigma}^{\tau}$ is proposed here in context of algebraic analysis \cite{DPR}. The linear operator $D_{\sigma}^{\tau}$, specified with the help of two fixed maps…
Suppose that $X\_{1}$ and $X\_{2}$ are two selfadjoint random variables that are freely independent over an operator algebra $\mathcal{B}$. We describe the possible operator atoms of the distribution of $X\_{1}+X\_{2}$ and, using…
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a $\operatorname{D}$-Lie algebra. A $\operatorname{D}$-Lie algebra $\tilde{L}$…
The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are…
We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes.…
We give a spectral theorem for unital representations of Hermitian commutative unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A better result is known for the case in which the *-algebra is countably generated.
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…
In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation $\mathcal{E}$ are understood as B\"{a}cklund auto-transformations of the equation $\mathcal{TE}$ tangent to $\mathcal{E}$. We…
We consider pairs of operators $A,B\in B(H)$, where $H$ is a Hilbert space, such that there exist a linear isometry $f$ from the span of $\{A,B\}$ into $\mathbb{C}^2$ mapping $A,B$ into orthonormal vectors. We prove some necessary…