Related papers: The Dirac operator of a commuting d-tuple

Let $T = (T_1, \ldots, T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an…

Let $\mathcal U(d)$ be the group of $d\times d$ unitary matrices. We find conditions to ensure that a $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol T$ is unitarily equivalent to multiplication by the coordinate functions on some…

Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or…

This article studies a class of Dirac operators of the form $D_\varepsilon= D+\varepsilon^{-1}\mathcal A$, where $\mathcal A$ is a zeroth order perturbation vanishing on a subbundle. When $\mathcal A$ satisfies certain additional…

We give more details about an integrable system in which the Dirac operator D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) = d(t)…

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice of a…

An \textit{ideal} of $N$-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with…

In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define…

The one-dimensional Dirac operator with periodic potential $V=\begin{pmatrix} 0 & \mathcal{P}(x) \\ \mathcal{Q}(x) & 0 \end{pmatrix}$, where $\mathcal{P},\mathcal{Q}\in L^2([0,\pi])$ subject to periodic, antiperiodic or a general strictly…

In this paper we prove that Dirac operators on non-compact complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes'/Divergence theorem.

R. W. Carey and J. Pincus in [CaPi86] proposed and index theory for non-Fredholm bounded operators T on a separable Hilbert space H such that TT* - T*T is in the trace class. We showed in [CGK13] using Dirac-type operators acting on…

We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular…

By a theorem of Dixmier-Douady the unitary group of an infinite-dimensional separable Hilbert space $H$ in the strong operator topology is contractible. The Dixmier-Douady proof is based on an explicit construction of families of subspaces…

A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant…

We show that even if a lattice Dirac operator satisfies the conditions consisting of locality, free of species doublings, correct continuum behavior, $\gm5$-hermiticity and the Ginsparg-Wilson relation, it does not necessarily have exact…

The tetrad gauge invariant theory of the free Dirac field in two special moving charts of the de Sitter spacetime is investigated pointing out the operators that commute with the Dirac one. These are the generators of the symmetry…

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides…

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we…

This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, which is a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther [Bull.…

We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra $\mathcal{W}_q(\mathfrak{sl}_2)$ for $U_q(\mathfrak{sl}_2)$ and the…