The Functional Calculus for Sectorial Operators

Haase, Markus

doi:10.1007/3-7643-7698-8

Cited by 481 publications

(473 citation statements)

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“…If −A generates a uniformly bounded C 0 -semigroup, then the Hille-Phillips calculus extends the holomorphic functional calculus for angles ψ ∈ ( π 2 , π), see Lemma 3.3.1 and Proposition 3.3.2 in [6]. In particular, for α ∈ C + the fractional power A α can be defined in the Hille-Phillips calculus yielding the same operator as in the holomorphic functional calculus.…”

Section: Semigroupsmentioning

confidence: 97%

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Convergence of subdiagonal Padé approximations of C 0-semigroups

Egert

Rozendaal

2013

J. Evol. Equ.

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Abstract. Let (rn) n∈N be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for −A the generator of a uniformly bounded C 0 -semigroup T on a Banach space X, the sequence (rn(−tA)) n∈N converges strongly to T (t) on D(A α ) for α > 1 2 . Local uniform convergence in t and explicit convergence rates in n are established. For specific classes of semigroups, such as bounded analytic or exponentially γ-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.

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Section: Semigroupsmentioning

confidence: 97%

“…If A is a sectorial operator of angle ϕ ∈ (0, π), the holomorphic functional calculus for A is defined, according to [6,Ch. 2], as follows.…”

Section: Semigroupsmentioning

confidence: 99%

Convergence of subdiagonal Padé approximations of C 0-semigroups

Egert

Rozendaal

2013

J. Evol. Equ.

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“…For a recent reference including several examples and properties we refer the reader to [74]. Note that an operator is sectorial of type if and only if − is sectorial of type 0.…”

Section: Definition 16 ([74] Sectorial Operator) a Closed And Linearmentioning

confidence: 99%

“…Very recently, Cuesta in [74](Theorem 1) has proved that if is a sectorial operator of type < 0 for some > 0 and 0 ≤ < (1 − /2), then there exists > 0 such that…”

Section: Definition 16 ([74] Sectorial Operator) a Closed And Linearmentioning

confidence: 99%

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

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This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations D ( ) = ( ) + D −1 ( , ( ), ( )), ∈ R, where 1 < < 2, is a linear densely defined operator of sectorial type on a complex Banach space and is a bounded linear operator defined on , is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

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“…In other words, operators such as H c are still sectorial in the sense of [Haa06]. In particular, there exists an analytic functional calculus for these operators which, in turn, yields the existence e.g.…”

Section: Counterexample: An Operator With Negative Real Partmentioning

confidence: 99%

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

Dondl

Dorey

Rösler

2016

Appl Math Res Express

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We are concerned with the non-normal Schrödinger operatorThe spectrum of this operator is discrete and contained in the positive half plane. In general, the ε-pseudospectrum of H will have an unbounded component for any ε > 0 and thus will not approximate the spectrum in a global sense.By exploiting the fact that the semigroup e −tH is immediately compact, we show a complementary result, namely that for every δ > 0, R > 0 there exists an ε > 0 such that the ε-pseudospectrum{z : |z − λ| < δ}.In particular, the unbounded part of the pseudospectrum escapes towards +∞ as ε decreases.Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.Mathematics Subject Classification (2010): 35P99, 47B44.

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The Functional Calculus for Sectorial Operators

Cited by 481 publications

References 0 publications

Convergence of subdiagonal Padé approximations of C 0-semigroups

Convergence of subdiagonal Padé approximations of C 0-semigroups

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

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