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Let $\nu$ be a finite complex measure with support in $\bar {\mathbb D}$ and let $\mathcal C\nu$ denote the Cauchy transform of $\nu .$ Suppose that $\nu$ annihilates polynomials in complex variable $z$ and $\nu |_{\partial \mathbb D} =…

We obtain sharp approximation results for into nearisometries between Lp spaces and nearisometries into a Hilbert space. Our main theorem is the optimal approximation result for nearsurjective nearisometries between general Banach spaces.

By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d.\ sequences. This observation not only explains the remarkable properties of lacunary trigonometric…

We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this…

We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,{\tau})-approximation property if E-valued operators belonging to…

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in\mathcal{L}(X)$ admits an…

Given a family of subspaces we investigate existence, quantity and quality of common complements in Hilbert spaces and Banach spaces. In particular we are interested in complements for countable families of closed subspaces of finite…

Given a homogeneous component of an exterior algebra, we characterize those subspaces in which every nonzero element is decomposable. In geometric terms, this corresponds to characterizing the projective linear subvarieties of the Grassmann…

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry \`a la Hitchin). The generalized twistor space associated to such a manifold is…

We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces

We consider ill-posed linear operator equations with operators acting between Banach spaces. For solution approximation, the methods of choice here are projection methods onto finite dimensional subspaces, thus extending existing results…

Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to…

We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of $\mathbb{C}^n$ in terms of the spectrum of both the unperturbed \& perturbed matrices, as well…

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on $\mathbb{R}$. We show that spectral multiplicity has a uniform…

We show that every metric space with bounded geometry uniformly embeds into an explicit reflexive Banach space (a direct sum of l^p spaces). In the case of discrete groups we show the analogue of a-T-menability. That is, we construct a…

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group $G_S(k)$ of the maximal extension of a global…

For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for…

Gaussian random matrix ensembles defined over the tangent spaces of the large families of Cartan's symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics since they describe the universal ergodic limit of…

We show that for any separable reflexive Banach space $X$ and a large class of Banach spaces $E$, including those with a subsymmetric shrinking basis but also all spaces $L_p$ for $1\leq p \leq \infty$, every bounded linear map ${\mathcal…

It is now well known that sparse or compressible vectors can be stably recovered from their low-dimensional projection, provided the projection matrix satisfies a Restricted Isometry Property (RIP). We establish new implications of the RIP…