Some results on spectra and certain norms
Danielle Witt
TL;DR
The paper investigates which spectra (and joint spectra) can arise from given norms of powers in Banach algebras. It extends the classical spectral radius formula to the joint setting, identifying the smallest polynomially convex circled hull containing the joint spectrum in terms of monomial norms, and proves that this hull is the maximal realizable joint spectrum under those norms. It also develops constructive methods (via weighted shifts) to realize prescribed norm data and to determine the largest rationally convex circled sets contained in, or equal to, the joint spectrum hull, along with lower bounds on monomial norms over the spectrum. The results connect spectral geometry with algebraic hulls (polynomially and rationally convex circled sets) and offer open questions about attainability of certain hulls as actual joint spectra.
Abstract
Given the norms of powers $(\lVert x^n\rVert)_{n\geq 0}$ of a Banach algebra element $x$, the largest possible value of the minimum modulus on the spectrum of $x$ is determined. It is also shown that, given a Banach algebra element $x$ and a compact set $K\subset\mathbb{C}$ with maximum modulus no more than the spectral radius of $x$, there exists a Banach algebra element $y$ with $\lVert y^n\rVert=\lVert x^n\rVert$ for all $n\geq 0$ and spectrum equal to the union of the spectrum of $x$ and $K$. These results, along with the spectral radius formula, are generalized to the joint spectrum of several commutative Banach algebra elements. The generalization of the spectral radius formula presented gives the maximum possible joint spectrum for commutative Banach algebra elements $x_1,\ldots,x_n$, given the norms $(\lVert x_1^{i_1}\cdots x_n^{i_n}\rVert)_{i_1,\ldots,i_n\geq 0}$.