On numerical ranges and roots.
Let \({\mathfrak a}\) be a complex Banach algebra with unit \(e\). The algebraic numerical range \(V(a)\) of an element \(a\) in \({\mathfrak a}\) is the set \(\{f(a): f\) is a bounded linear functional on \({\mathfrak a}\) such that \(f(e)= \| f\|=1\}\). It is shown that if \(V(a)\) contains no negative real number, then for any \(w\), \(0< w< 1\), there exists a \(b_w\), the \(w\)th power of \(a\), in the closed unital subalgebra of \({\mathfrak a}\) generated by \(a\) whose spectrum is contained in the wedge \(\{re^{i\theta}\in \mathbb{C}: r\geq 0\), \(|\theta|< w\pi\}\) such that \(b_{w_1} b_{w_2}= a\) if \(w_1+w_2=1\). It is unique if \(a\) is invertible and \(w\) is the reciprocal of an integer. For a Hermitian Banach *-algebra \({\mathfrak a}\) (a Banach algebra with a conjugate linear involution * satisfying \((xy)^*= y^*x^*\) for all \(x\) and \(y\) such that every Hermitian element has real spectrum), the numerical range \(V(a)\) should be replaced by the *-numerical range \(V_*(a)= \{f(a): f\) as before with \((fx^*x)\geq 0\) for all \(x\) in \({\mathfrak a}\}\). If \({\mathfrak a}\) further satisfies the condition that \(V_*(a)\) contains \(\sigma(a)\) for every \(a\) in \({\mathfrak a}\), then it is proved that under the previous assumptions on \(a\) and \(w\), there exists \(b_w\) such that \(V_*(b_w)\) is itself contained in the wedge \(\{re^{i\theta}: r\geq 0\), \(|\theta|\leq w\pi\}\). These results generalize the corresponding ones for operators on a Hilbert space.
scientific article; zbMATH DE number 1956920
scientific article