Weakly admissible H-calculus on reflexive Banach spaces

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Abstract

We show that, given a reflexive Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction.
Original languageEnglish
Pages (from-to)796-815
Number of pages20
JournalIndagationes mathematicae
Volume23
Issue number4
DOIs
Publication statusPublished - Dec 2012

Fingerprint

Bounded Analytic Functions
Functional Calculus
Reflexive Banach Space
Separable Hilbert Space
Toeplitz Operator
Admissibility
Observability
Systems Theory
Half-plane
Calculus
Semigroup
Generator
Operator

Keywords

  • MSC-47A60
  • MSC-47B35
  • MSC-47D06
  • MSC-93C25
  • IR-82456
  • Weak admissibility
  • Operator semigroup
  • EWI-22631
  • METIS-296163
  • Functional calculus

Cite this

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title = "Weakly admissible H-calculus on reflexive Banach spaces",
abstract = "We show that, given a reflexive Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction.",
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Weakly admissible H-calculus on reflexive Banach spaces. / Schwenninger, F.L.; Zwart, Heiko J.

In: Indagationes mathematicae, Vol. 23, No. 4, 12.2012, p. 796-815.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Weakly admissible H-calculus on reflexive Banach spaces

AU - Schwenninger, F.L.

AU - Zwart, Heiko J.

PY - 2012/12

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N2 - We show that, given a reflexive Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction.

AB - We show that, given a reflexive Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction.

KW - MSC-47A60

KW - MSC-47B35

KW - MSC-47D06

KW - MSC-93C25

KW - IR-82456

KW - Weak admissibility

KW - Operator semigroup

KW - EWI-22631

KW - METIS-296163

KW - Functional calculus

U2 - 10.1016/j.indag.2012.04.005

DO - 10.1016/j.indag.2012.04.005

M3 - Article

VL - 23

SP - 796

EP - 815

JO - Indagationes mathematicae

JF - Indagationes mathematicae

SN - 0019-3577

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ER -