Abstract
Notation. Throughout this chapter, U,X and Y are complex Hilbert spaces which are identified with their duals. \( \mathbb{T} \) is a strongly continuous semigroup on X, with generator A: D(A) → X and growth bound ω0(\( \mathbb{T} \)). Recall from Section 2.10 that X1 is D(A) with the norm ‖z‖1 = ‖(βI - A)z‖, where β∋ρ(A) is fixed, while X−1 is the completion of X with respect to the norm ‖z‖−1 = ‖ (βI - A)−1z‖. Remember that we use the notation A and \( \mathbb{T}_t \) also for the extension of the original generator to X and for the extension of the original semigroup to X−1. Recall also that X 1 d is D(A*) with the norm \( ||z||_1^d = ||(\bar \beta I - A*)z|| \) and X −1 d is the completion of X with respect to the norm \( ||z||_{ - 1}^d = ||(\bar \beta I - A*)^{ - 1} z|| \) . Recall that X−1 is the dual of X 1 d with respect to the pivot space X.
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© 2009 Birkhäuser Verlag AG
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(2009). Control and Observation Operators. In: Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts / Basler Lehrbücher. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8994-9_4
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DOI: https://doi.org/10.1007/978-3-7643-8994-9_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8993-2
Online ISBN: 978-3-7643-8994-9
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