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Michel Crouzeix

Publications and source records attributed to Michel Crouzeix.

17 recordsLinked to original sources
arXiv2025

Numerical Ranges and Spectral Sets: the unbounded case

It is known that, if $Ω$ $\subset$ C is a convex set containing the numerical range of an operator A, then $Ω$ is a C $Ω$ -spectral set for A with C $Ω$ $\le$ 1+ $\sqrt$ 2. We improve this estimate in unbounded cases.

math.FA↗
arXiv2020

A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions

A result by Crouzeix and Palencia states that the spectral norm of a matrix function $f(A)$ is bounded by $K = 1+\sqrt{2}$ times the maximum of $f$ on $W(A)$, the numerical range of $A$. The purpose of this work is to point out that this result extends to a certain notion of bivariate matrix functions; the spectral norm of $f\{A,B\}$ is bounded by $K^2$ times the maximum of $f$ on $W(A)\times W(B)$. As a special case, it follows that the spectral norm of the Fréchet derivative of $f(A)$ is bounded by $K^2$ times the maximum of $f^\prime$ on $W(A)$. An application to the convergence analysis of certain Krylov subspace methods and the extension to functions in more than two variables are discussed.

math.FA↗
arXiv2018

Spectral sets and operator radii

We study different operator radii of homomorphisms from an operator algebra into $B(H)$ and show that these can be computed explicitly in terms of the usual norm. As an application, we show that if $Ω$ is a $K$-spectral set for a Hilbert space operator, then it is a $M$-numerical radius set, where $M=\frac{1}{2}(K+K^{-1})$. This is a counterpart of a recent result of Davidson, Paulsen and Woerdeman. More general results for operator radii associated with the class of operators having $ρ$-dilations in the sense of Sz.-Nagy and Foias are given. A version of a result of Drury concerning the joint numerical radius of non-commuting $n$-tuples of operators is also obtained.

math.FA↗
arXiv2018

Spectral Sets: Numerical Range and Beyond

We extend the proof in [M.~Crouzeix and C.~Palencia, {\em The numerical range is a $(1 + \sqrt{2})$-spectral set}, SIAM Jour.~Matrix Anal.~Appl., 38 (2017), pp.~649-655] to show that other regions in the complex plane are $K$-spectral sets. In particular, we show that various annular regions are $(1 + \sqrt{2} )$-spectral sets and that a more general convex region with a circular hole or cutout is a $(3 + 2 \sqrt{3} )$-spectral set. We demonstrate how these results can be used to give bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of rational Krylov subspace methods for approximating $f(A)b$, where $A$ is a square matrix, $b$ is a given vector, and $f$ is a function that can be uniformly approximated on such a region by rational functions with poles outside the region.

math.SP↗
arXiv2017

The numerical range as a spectral set

It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) $(1{+}\sqrt2)$-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of holomorphic functions.

math.FA↗
arXiv2016

Some constants related to numerical ranges

In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2--spectral set for the matrix A, we propose a study of various constants. We review some partial results, many problems are still open. We describe our corresponding numerical tests.

math.FA↗
arXiv2014

The inf-sup constant for the divergence on corner domains

The inf-sup constant for the divergence, or LBB constant, is related to the Cosserat spectrum. It has been known for a long time that on non-smooth domains the Cosserat operator has a non-trivial essential spectrum, which can be used to bound the LBB constant from above. We prove that the essential spectrum on a plane polygon consists of an interval related to the corner angles and that on three-dimensional domains with edges, the essential spectrum contains such an interval. We obtain some numerical evidence for the extent of the essential spectrum on domains with axisymmetric conical points by computing the roots of explicitly given holomorphic functions related to the corner Mellin symbol. Using finite element discretizations of the Stokes problem, we present numerical results pertaining to the question of the existence of eigenvalues below the essential spectrum on rectangles and cuboids.

math.NA↗
arXiv2013

Faber polynomials of matrices for non-convex sets

It has been recently shown that $|| F_n(A) ||\leq 2$, where $A$ is a linear continuous operator acting in a Hilbert space, and $F_n$ is the Faber polynomial of degree $n$ corresponding to some convex compact $E\subset \mathbb C$ containing the numerical range of $A$. Such an inequality is useful in numerical linear algebra, it allows for instance to derive error bounds for Krylov subspace methods. In the present paper we extend this result to not necessary convex sets $E$.

math.NA↗
arXiv2012

Numerical radius and distance from unitary operators

Denote by w(A) the numerical radius of a bounded linear operator A acting on Hilbert space. Suppose that A is invertible and that the numerical radius of A and of its inverse are no greater than 1+e for some non-negative e. It is shown that the distance of A from unitary operators is less or equal than a constant times $e^{1/4}$. This generalizes a result due to J.G. Stampfli, which is obtained for e = 0. An example is given showing that the exponent 1/4 is optimal. The more general case of the operator $ρ$-radius is discussed for $ρ$ between 1 and 2.

math.FA↗
arXiv2011

Energy of N Cooper pair by analytically solving Richardson-Gaudin equations

This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these $N$ pairs via $N$ parameters coupled through $N$ non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the $N$-pair energy depends on $N$ as $N(N-1)$ only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.

cond-mat.supr-con↗
arXiv2008

Intersections of several disks of the Riemann sphere as K-spectral sets

We prove that if $n$ closed disks $D_1, D_2, ..., D_n$, of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_1\cap D_2...\cap D_n$ is a complete $K$-spectral set for $A$, with $K\leq n+n(n-1)/\sqrt3$. When $n=2$ and the intersection $X_1\cap X_2$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).

math.SP↗
arXiv2007

K-spectral sets and intersections of disks of the Riemann sphere

We prove that if two closed disks X_1 and X_2 of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then the intersection X_1\cap X_2 is a complete (2+2/\sqrt{3})-spectral set for A. When the intersection of X_1 and X_2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).

math.SP↗
arXiv2005

Convex domains and K-spectral sets

Let $Ω$ be an open convex domain of the complex plane. We study constants K such that $Ω$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $Ω$. Several approaches are discussed.

math.FA↗