On the abstract theorem of Picard
The result named in the title states that if \(A\) is a complex unital Banach algebra, \(J\) a two-sided closed ideal and \(D:A\rightarrow A\) a continuous derivation, then for any \(a\in A\) with \(Da\not\in J\) we have \[ \bigcup\{V((\exp \lambda D_J)({\widetilde a})):\lambda\in {\mathbf C}\}={\mathbf C}. \] Here \(V\) denotes the numerical range, \(D_J\) is the induced derivation on \(A/J\) and \({\widetilde a}\) is the coset of \(a\) in \(A/J\). Other results, of a rather technical nature, are also obtained.
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