Implicit gamma theorems. I: Pseudoroots and pseudospectra (Q1396183)

Consider the analytic map \(g:E\rightarrow F\) between two Hilbert spaces, where \(g(x)\) is the solution of an equation which depends on the parameter \(x\). The authors study the \(\varepsilon\)-pseudosolutions, \(g\left( B\left( x,\varepsilon \right) \right) \), where \(B\left( x,\varepsilon \right) \) is a ball about \(x\) and of radius \(\varepsilon\). The objective here is to estimate the size of the set \(g\left( B\left( x,\varepsilon \right) \right) \) by using the derivative \(Dg(x).\) The results are applied to the implicit function map and to the polynomial eigenvalue problem.