In this paper, we investigate the numerical ranges of composition operators, whose symbols are elliptic automorphisms finite orders on Hilbert–Hardy space $$H^2(D).$$ We give complete descriptions operators induced by order 2 and 3.

Kippenhahn's Theorem asserts that the numerical range of a matrix is convex hull certain algebraic curve. Here, we show joint finitely many Hermitian matrices similarly semi-algebraic set. We discuss an analogous statement regarding dual cone to hyperbolicity and prove class bases these cones closed under linear operations. The result offers new geometric method analyze quantum states.