Analytic perturbation of the Taylor spectrum

Abstract:

Let ${T_1}(z), \ldots ,{T_m}(z)$, $z \in G \subset {{\mathbf {C}}^k}$, be analytic families of bounded operators in a complex Banach space $X$, such that for each $z \in G$ the operators ${T_i}(z)$ and ${T_j}(z)$, $i,j = 1, \ldots ,n$, commute. Main result: If $K(z)$ denotes the Taylor spectrum of the tuple $({T_1}(z), \ldots ,{T_m}(z))$, then the set-valued function $K:G \to {2^{{\mathbf {C}}m}}$ is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of $k$-tuples of complex polynomials on the graph of $K$.