Abstract
Since |(x, y)| = |(A−s/2x,As/2y)| ≤ ‖x‖−s · ‖y‖s for all x ∈ H and y ∈ Hs, s > 0, it follows that the sesquilinear form (x, y) extends from H ×Hs by continuity to a sesquilinear form (x, y)−s,s onH−s ×Hs. Consequently,H−s can be treated as the dual space of Hs (H−s = H∗ s ); the duality pairing is given by the sesquilinear form (x, y)−s,s (duality with respect toH). ByL (X,Y ) we denote the set of bounded linear operators from a spaceX to a space Y . The operator A can be viewed as a bounded operator in a scale of Hilbert spaces, A ∈ L (Hs,Hs−2) for s ∈ R. Just as in [3], we make the following assumption about the operatorD.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.