The Taylor Spectrum of Pairs of Isometries

In the paper we fully describe the Taylor spectrum of pairs of isometries defined by diagrams. In most cases both isometries in such pairs have nontrivial shift part, and its Taylor spectrum is a proper subset (of Lebesgue measure in $$(0,\pi ^2)$$ ) of the closed bidisk. In particular, there are pairs of isometries which spectrum is equal to $$\{(\mu ,\lambda )\in \overline{\mathbb {D}}^2: |\mu |^a\le |\lambda |\le |\mu |^b\}$$ for any $$0<a\le b<\infty $$ .