Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-qn7h5 Total loading time: 0.286 Render date: 2022-10-04T06:34:11.432Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": true, "useSa": true } hasContentIssue true

10 - Resolvable Input/State/Output and State/Signal Nodes

Published online by Cambridge University Press:  05 May 2022

Damir Z. Arov
Affiliation:
South Ukrainian Pedagogical University
Olof J. Staffans
Affiliation:
Åbo Akademi University, Finland
Get access

Summary

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).

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×