On the C∞ Chevalley's theorem

Dadok, Jiri

doi:10.1016/0001-8708(82)90002-0

Cited by 24 publications

(19 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following result, however, is less obvious. In this form it was proved directly by Dadok [5]; we give a different (though related) proof which we then use to extend the result to the appropriate compactifications. Proof.…”

Section: Invariant Smooth Functionsmentioning

confidence: 84%

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Mazzeo

Vasy

2005

Journal of Functional Analysis

View full text Add to dashboard Cite

Let (M, g) be a globally symmetric space of noncompact type, of arbitrary rank, and its Laplacian. We introduce a new method to analyze and the resolvent ( − ) −1 ; this has origins in quantum N-body scattering, but is independent of the 'classical' theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.

show abstract

Section: Invariant Smooth Functionsmentioning

confidence: 84%

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Mazzeo

Vasy

2005

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…Analogues exist for C (∞) and analytic functions-see [19]. At the opposite extreme, at least for spectral functions, we have the following result (see [37,46]).…”

Section: Theorem 56 (Subdifferentials Of Invariant Functions) In Thmentioning

confidence: 61%

The mathematics of eigenvalue optimization

Lewis

2003

Math. Program., Ser. B

View full text Add to dashboard Cite

Abstract. Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.Key words. Eigenvalue optimization -convexity -nonsmooth analysis -duality -semidefinite program -subdifferential -Clarke regular -chain rule -sensitivity -eigenvalue perturbation -partly smooth -spectral function -unitarily invariant norm -hyperbolic polynomial -stability -robust control -pseudospectrum -H∞ norm PART I: INTRODUCTION Von Neumann and invariant matrix normsBefore outlining this survey, I would like to suggest its flavour with a celebrated classical result. This result, von Neumann's characterization of unitarily invariant matrix norms, serves both as a historical jumping-off point and as an elegant juxtaposition of the central ingredients of this article.Von Neumann [64] was interested in unitarily invariant norms · on the vector space M n of n-by-n complex matrices:where U n denotes the group of unitary matrices. The singular value decomposition shows that the invariants of a matrix X under unitary transformations of the form X → U XV are given by the singular values σ 1 (X) ≥ σ 2 (X) ≥ · · · ≥ σ n (X), the eigenvalues of the matrix √ X * X. Hence any invariant norm · must be a function of the vector σ(X), so we can write X = g(σ(X)) for some function g : R n → R. We ensure this last equation if we define g(x) = Diag x for vectors x ∈ R n , and in that case g is a symmetric gauge function: that is, g is a norm on R n whose value is invariant under permutations and sign changes of the components. Von Neumann's beautiful insight was that this simple necessary condition is in fact also sufficient.Theorem 1.1 (von Neumann, 1937). The unitarily invariant matrix norms are exactly the symmetric gauge functions of the singular values.Von Neumann's proof rivals the result in elegance. He proceeds by calculating the dual norm of a matrix Y ∈ M n ,where the real inner product X, Y in M n is the real part of the trace of X * Y . Specifically, he shows that any symmetric gauge function g satisfies the simple duality relationshipwhere g * is the norm on R n dual to g. From this, the hard part of his result follows:...

show abstract

“…Thus they extend to an H -invariant scalar product and norm q 0 and q 1 with q = q 0 + q 1 . Finally, an H -invariant function is smooth if and only if its restriction to W is smooth [2]. Smoothingq at the origin, we deduce, thatq is smooth in \{0} if and only if q is smooth in V \{0}.…”

Section: Proposition 44 the Restriction Q →Q Is A Bijection Between mentioning

confidence: 76%