Related papers: Operator-Lipschitz functions in Schatten-von Neuma…

We study the optimal sets $\Omega^\ast\subset\mathbb{R}^d$ for spectral functionals $F\big(\lambda_1(\Omega),\dots,\lambda_p(\Omega)\big)$, which are bi-Lipschitz with respect to each of the eigenvalues…

We give a proof that the first eigenfunction of the $\alpha$-symmetric stable process on a bounded Lipschitz domain in $\R^d$, $d\geq 1$, is superharmonic for $\alpha=2/m$, where $m>2$ is an integer. This result was first proved for the…

This paper is dedicated to the spectral optimization problem \begin{equation*} \min \big\{ \lambda_1(\Omega)+\cdots+\lambda_k(\Omega) + \Lambda|\Omega| \ : \ \Omega \subset D \text{ quasi-open} \big\} \end{equation*} where…

We establish, for $1 < p < \infty$, higher order $\mathcal{S}^p$-differentiability results of the function $\varphi : t\in \mathbb{R} \mapsto f(A+tK) - f(A)$ for selfadjoint operators $A$ and $K$ on a separable Hilbert space $\mathcal{H}$…

Suppose $\alpha$ is an orientation preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. We establish sufficient conditions for the Fredholmness of the singular integral operator \[…

We study linear perturbations of Donoghue classes of scalar Herglotz-Nevanlinna functions by a real parameter $Q$ and their representations as impedance of conservative L-systems. Perturbation classes $\mathfrak M^Q$, $\mathfrak…

We study composition operators on spaces of holomorphic Lipschitz functions defined on the open unit ball of a complex Banach space. Our approach is based on the linearization of the symbol through the holomorphic Lipschitz-free spaces,…

The theory of abstract Friedrichs operators was introduced some fifteen years ago with the aim of providing a more comprehensive framework for the study of positive symmetric systems of first-order partial differential equations, nowadays…

We prove that the potential of a Sturm--Liouville operator depends analytically and Lipschitz continuously on the spectral data (two spectra or one spectrum and the corresponding norming constants). We treat the class of operators with…

The non-Leibniz formalism is introduced in this article. The formalism is based on the generalized differentiation operator (kappa-operator) with a non-zero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on…

Let $H$ be a real Hilbert space and $\Phi:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $\Phi^{-1}(0)\neq \emptyset$ if and only if, for each $\epsilon>0$, there exist a convex set $X\subset H$ and…

We prove a general Levinson's theorem for Schr\"odinger operators in two dimensions with threshold obstructions at zero energy. Our results confirm and simplify earlier seminal results of Boll\'e, Gesztesy et al., while providing an…

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as…

In this paper, we revisit the Riemann--Liouville analytic semigroup. In particular, we completely characterize the membership to the Schatten class $S^r$ on $L^2(0,1)$, as well as the membership to the class of nuclear operators on…

We consider rank one perturbations $A_\alpha=A+\alpha(\cdot,\varphi)\varphi$ of a self-adjoint operator $A$ with cyclic vector $\varphi\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed…

We study the spectral minimum and Lifshitz tails for continuum random Schr\"{o}dinger operators of the form \begin{equation*} H_{\om}=-\De+V_{0}+\sum_{i\in\Z^{d}}\om_{i}u(\cdot-i), \end{equation*} where $V_{0}$ is the periodic potential,…

We treat the general theory of nonlinear ideals and extend as many notions as possible from the linear theory to the nonlinear theory. We define nonlinear ideals with special properties which associate new non-linear ideals to given ones…

In this work we study Schatten-von Neumann classes of tensor products of invariant operators on Hilbert spaces. In the first part we first deduce some spectral properties for tensors of anharmonic oscillators thanks to the knowledge on…

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to $\sigma\in…

We establish continuity and Schatten-von Neumann properties for Fourier integral operators with amplitudes in Orlicz modulation spaces, when acting on other Orlicz modulation spaces themselves. The phase functions are non smooth and admit…