Projective spectrum and kernel bundle. II

In this paper, we study the projective joint spectrum P(A) and P(A*) of the operator tuple A = (A₁, A₂, . . ., A_n). We first compute the joint spectrum for the Cuntz tuple. Then we study tuples of compact operators on anWinfinite dimensional Banach space. We show that if P(A*) is smooth, then z2P(A*) ker A*(z) forms a holomorphic line bundle over P(A*). For linearly independent vectors e₁, e₂, e₃ and A_i = e_i ⊗ e_i, i = 1, 2, 3, the smoothness of P(A*) depends rather subtly on the relative position of the vectors. As an example, we compute the Chern character of the line bundle in the two vector case and show that it is nontrivial.