Published online by Cambridge University Press: 05 May 2022
In this chapter, we continue the study of the resolvent set and the i/s/o resolvent matrix of an i/s/o node Σ begun in Chapter 5. In particular, we show that if Σ is resolvable, i.e., if Σ has a nonempty resolvent set ρ(Σ), then the main operator A of Σ is also resolvable and ρ(Σ) = ρ(A). Moreover, the i/s/o resolvent matrix is analytic and satisfies the i/s/o resolvent identity in ρ(Σ). Even more interesting is the converse claim: every i/s/o pseudoresolvent is a restriction of the i/s/o resolvent matrix of a unique i/s/o node Σ, where we by an i/s/o pseudo-resolvent mean a locally bounded block matrix operator-valued function that satisfies the i/s/o resolvent identity in some open subset Ω of C. In particular, every i/s/o pseudo-resolvent is analytic. Our class of regular resolvable i/s/o nodes is known from before in the literature with more complicated definitions and different names (e.g., in Staffans, 2005; systems that belong to this class are called “operator nodes”). At the end of this chapter, we continue the study of the connection between the characteristic bundles of a s/s system Σ and the i/s/o resolvent matrices of i/s/o representations of Σ and show that these characteristic bundles are analytic in ρ(Σ).