Related papers: On the Index of a Non-Fredholm Model Operator

We study an analogue of chirality operators associated with quantum walks on the binary tree. For those operators we introduce a K-theoretic invariant, an analogue of the index of Fredholm operators, and compute its values in the ring of…

Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$…

We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in…

While in \cite{HB} we studied classes of Fredholm-type operators defined by the homomorphism $\Pi$ from $L(X)$ onto the Calkin algebra $\mathcal{C}(X)$, $X$ being a Banach space, we study in this paper two classes of Fredholm-type operators…

In the present work we consider in $L^2(\mathbb{R}_+)$ the Schr\"odinger operator $\mathrm{H_{X,\alpha}}=-\mathrm{\frac{d^2}{dx^2}}+\sum_{n=1}^{\infty}\alpha_n\delta(x-x_n)$. We investigate and complete the conditions of self-adjointness…

Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1\pi\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint.…

We consider the $3-D$ Dirac operator $\mathfrak{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ with variable regular magnetic and electrostatic potentials $ \boldsymbol{A}$,$\Phi $ and with singular potentials $Q_{\sin }$ with support on a smooth…

We use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a real-analytic path having possibly non transversal…

We show that a recent spectral flow approach proposed by Berkolaiko-Cox-Marzuola for analyzing the nodal deficiency of the nodal partition associated to an eigenfunction can be extended to more general partitions. To be more precise, we…

We introduce a family of differential-reflection operators $\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For…

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(\Omega; d^n x)$, $n\geq 2$, where $\Omega$ is an open set with a compact, nonempty boundary…

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a H\"older space with respect to the spatial variable. As a main ingredient for the proof we use a…

We consider an action of the real line on a C*-algebra for which there is a centre-valued invariant trace. We define a family of Toeplitz operators with symbols in the original algebra. When the symbol is invertible, the Toeplitz operator…

We present a natural imbedding of the crossed product $\mathcal A \rtimes_\xi \mathbb Z^d$ into the $C^\ast$-algebra of adjointable operators over the standard Hilbert $\mathcal A$-module $\mathcal H_{\mathcal A}$. By replacing the…

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift…

In the first part of the article we establish the existence in the sense of sequences of solutions in $H^{2}(R)$ for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is…

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. We assume that the unperturbed operator has a periodic Hamilton flow,…

Let $\mathcal{A}$ denote the operator class in which every nonzero intertwiner between two operators in $\mathcal{A}$ has dense range. Utilizing the operators in $\mathcal{A}$ as atoms and the flag structure as connection, we introduce an…

For trace class operators $A, B \in \mathcal{B}_1(\mathcal{H})$ ($\mathcal{H}$ a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form \[ {\det}_{\mathcal{H}} ((I_{\mathcal{H}} - A)…

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic…