Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra

Abstract. Let g : E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ E, 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 form:

g(x) + Dg (x) ( B(0, c ₁ ε N))

\subseteq

g(B(x,ε ))

\subseteq

g(x) +Dg (x) ( B(0, c ₂ ε ) ),

where Dg (x) denotes the derivative of g at x . We consider both the case where g is given explicitly and the case where g is given implicitly. We apply our results to the implicit function associated with the evaluation map, namely the solution map, and to the polynomial eigenvalue problem. Our results are stated in terms of an invariant γ which has been extensively used by various authors in the study of Newton's method. The main tool used here is an implicit γ theorem, which estimates the γ of an implicit function in terms of the γ of the function defining it.