A guess on hyponormal operator tuples (Q1200696)

The elementary conclusion that for a hyponormal operator \(T\) and a polynomial \(p\) the fact \(\sigma(p(T))=0\) implies \(p(T)=0\) is investigated -- according to a suggestion of R. E. Curto -- in the frame of tuples \((T_ 1,\dots,T_ n)\) of hyponormal operators: Does \(\sigma(p(T_ 1,\dots,T_ n))=\{0\}\) imply \(p(T_ 1,\dots,T_ n)=0\) for a polynomial \(p(.,\dots,.)\)? It turns out that a sufficient condition is the doubly commuting property.