On the numerical ranges of a family of commuting operators (Q1277588)

Let \(B\) be a complex Banach space. According to the Hahn-Banach theorem, for any element \(y\in B\) there exists a continuous linear functional (at least one) \(f_y\in B^*\) such that \(f_y(y)= \| y\|^2\) and \(\| f_y\|= \| y\|\). The choice of a functional \(f_y\) for each \(y\in B\) supplies \(B\) with a semi-inner product \([\cdot,\cdot]\) by the rule \([x,y]= f_y(x)\), \(x,y\in B\). Then by the numerical range of a bounded linear operator \(A\) on \(B\), we mean the set \(W(A):= \{[Ax,x]: x\in B\), \([x,x]=\| x\|^2=1\}\). The numerical range of an operator in a Banach space need not be convex, and generally depends on the choice of the functionals \(f_y\). Nevertheless, the convex hull \(\text{conv } \overline{W(A)}\) of the closure \(\overline{W(A)}\) is independent of the \(f_y\)'s that defines the semi-inner product on \(B\). The present note deals with the problem of modifying the numerical ranges of families of commuting operators under a modification of the original norm. The symbols \(\| A\|_*\) and \break \(W(A,\|\cdot \|_*)\) stand for the norm and the numerical range of an operator \(A\) corresponding to the norm \(\| \cdot\|_*\). Moreover, the inclusion \(\sigma(A) \subseteq \text{conv } \overline{W(A,\|\cdot\|_*)}\), \(\sigma(A)=\) spectrum of \(A\), establishes the smallest possible bound for the modification of the convex linear hull of the closure of the numerical range of an operator \(A\) under an equivalent renormalization \(\|\cdot \|_*\) of the Banach space. Simple examples show that the relation \(\text{conv } \overline{W(A, \|\cdot\|_*)}= \text{conv } \sigma(A)\) fails in general for an equivalent renormalization. Therefore, the following theorem is sharp in a certain sense. Theorem. Let \(A_1,\dots, A_n\) be pairwise commuting operators on a Banach space \(B\). Then for any \(\varepsilon >0\), there exists a norm \(\|\cdot \|_\varepsilon\) equivalent to the original norm such that the numerical range of each of the operators \(A_1,\dots, A_n\) belongs to the \(\varepsilon \)-neighborhood of the convex hull of the spectrum of this operator, i.e., \[ W(A_\gamma,\|\cdot \|_\varepsilon)\subset \{\lambda: | \lambda-\mu| < \varepsilon ,\;\varepsilon ,\mu\in\text{conv } \sigma(A_\gamma)\}, \quad \gamma= 1,\dots, n. \]