Generalizing Eigenvalue Theorems to Pseudospectra Theorems

Generalizing Eigenvalue Theorems to Pseudospectra Theorems

Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra

Let g: E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ F, the image under g of the closed ball about x ∈ E with radius ε. When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x, ε)) are ε-pseudosolutions. Our aim is to investigate the size of the set g(B(x, ε)). We derive upper and lower bounds of the following...

On Cheng’s Eigenvalue Comparison Theorems

We prove Cheng’s eigenvalue comparison theorems [4] for geodesic balls within the cut locus under weaker geometric hypothesis, Theorems (1.1, 3.1, 3.2) and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci curvature bounds. We construct examples of smooth m...

Generalizing Theorems in Real Closed Fields

Jan Kraj cek posed the following problem: Is there is a generalization result in the theory of real closed elds of the form: If A(1 + + 1) (n occurrences of 1) is provable in length k for all n 2 !, then (8x)A(x) is provable? It is argued that the answer to this question depends on the particular formulation of the \theory of real closed elds." Four distinct formulations are investigated with r...

Localization Theorems for Nonlinear Eigenvalue Problems

Abstract. Let T : Ω → C be a matrix-valued function that is analytic on some simplyconnected domain Ω ⊂ C. A point λ ∈ Ω is an eigenvalue if the matrix T (λ) is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three non...