The Functional Calculus for Sectorial Operators

Haase, Markus

doi:10.1007/3-7643-7698-8_2

Cited by 422 publications

(859 citation statements)

References 128 publications

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“…This functional calculus may be extended in a natural way in order to define the fractional powers

A^{ε}

for all

ε > 0

. It is known that

A^{ε}

is still a sectorial operator and

D false(A false) \subset D false(A^{ε} false)

when

0 < ε < 1

.…”

Section: Applicationsmentioning

confidence: 99%

“…Furthermore, A is said to admit a bounded

R H^{\infty}

‐functional calculus of angle

β \in true[ω_{H} (A), π true)

if, in addition, for all

β^{'} \in false(β, π false)

, the set

\begin{matrix} true \\ right true{f (A) : f \in H^{\infty} ({normalΣ}_{β^{'}}) and {false∥ f false∥}_{H^{\infty} ({normalΣ}_{β^{'}})} \leq 1 true} \end{matrix}

is R ‐bounded. We refer to for the concepts of

H^{\infty}

‐functional calculus and

R H^{\infty}

‐functional calculus for sectorial operators. Example Let

1 < p < \infty, \frac{1}{p} < α < 1

.…”

Section: Applicationsmentioning

confidence: 99%

“…For

k \in Z

, the operator

false(1 + a_{k} false) A + b_{k} A^{ε} - c_{k}

is a bijection from

D (A)

onto X and

{[false(1 + a_{k} false) A + b_{k} A^{ε} - c_{k}]}^{- 1} \in L false(X false)

as we have assumed that

ρ_{a, b} false(A, A^{ε} false) = Z

. One can verify, using for example [, Chapter 1], that

\begin{matrix} true \\ right f_{1} (A) = c_{k} {true[(1 + a_{k}) A + b_{k} A^{ε} - c_{k} true]}^{- 1}, f_{2} (A) = A^{ε} {true[(1 + a_{k}) A + b_{k} A^{ε} - c_{k} true]}^{- 1} \end{matrix}

when

false| k false| \geq k_{0}

. The assumption that A admits a bounded

R H^{\infty}

‐functional calculus of angle β implies that the sets

\begin{matrix} true \\ right true{c_{k} {false[(1 + a_{k}) A + b_{k} A^{ε} - c_{k} false]}^{- 1} : | k | \geq k_{0} true}, \end{matrix}

…”

Section: Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

Cai

2018

Mathematische Nachrichten

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We study the well-posedness of the fractional differential equations with infinite delayon Lebesgue-Bochner spaces ( ; ) and Besov spaces , ( ; ), where and are closed linear operators on a Banach space satisfying ( ) ∩ ( ) ≠ {0}, > 0 and , ∈ 1 (ℝ + ). Under suitable assumptions on the kernels and , we completely characterize the well-posedness of ( ) in the above vector-valued function spaces on by using known operator-valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied. K E Y W O R D SFourier multiplier, fractional differential equation, vector-valued function spaces, well-posedness M S C ( 2 0 1 0 ) 26A33, 34C25, 34K37, 43A15, 45N05

show abstract

“…This functional calculus may be extended in a natural way in order to define the fractional powers

A^{ε}

for all

ε > 0

. It is known that

A^{ε}

is still a sectorial operator and

D false(A false) \subset D false(A^{ε} false)

when

0 < ε < 1

.…”

Section: Applicationsmentioning

confidence: 99%

“…Furthermore, A is said to admit a bounded

R H^{\infty}

‐functional calculus of angle

β \in true[ω_{H} (A), π true)

if, in addition, for all

β^{'} \in false(β, π false)

, the set

\begin{matrix} true \\ right true{f (A) : f \in H^{\infty} ({normalΣ}_{β^{'}}) and {false∥ f false∥}_{H^{\infty} ({normalΣ}_{β^{'}})} \leq 1 true} \end{matrix}

is R ‐bounded. We refer to for the concepts of

H^{\infty}

‐functional calculus and

R H^{\infty}

‐functional calculus for sectorial operators. Example Let

1 < p < \infty, \frac{1}{p} < α < 1

.…”

Section: Applicationsmentioning

confidence: 99%

“…For

k \in Z

, the operator

false(1 + a_{k} false) A + b_{k} A^{ε} - c_{k}

is a bijection from

D (A)

onto X and

{[false(1 + a_{k} false) A + b_{k} A^{ε} - c_{k}]}^{- 1} \in L false(X false)

as we have assumed that

ρ_{a, b} false(A, A^{ε} false) = Z

. One can verify, using for example [, Chapter 1], that

\begin{matrix} true \\ right f_{1} (A) = c_{k} {true[(1 + a_{k}) A + b_{k} A^{ε} - c_{k} true]}^{- 1}, f_{2} (A) = A^{ε} {true[(1 + a_{k}) A + b_{k} A^{ε} - c_{k} true]}^{- 1} \end{matrix}

when

false| k false| \geq k_{0}

. The assumption that A admits a bounded

R H^{\infty}

‐functional calculus of angle β implies that the sets

\begin{matrix} true \\ right true{c_{k} {false[(1 + a_{k}) A + b_{k} A^{ε} - c_{k} false]}^{- 1} : | k | \geq k_{0} true}, \end{matrix}

…”

Section: Applicationsmentioning

confidence: 99%

See 1 more Smart Citation

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

Cai

2018

Mathematische Nachrichten

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show abstract

“…In this section we collect the main facts concerning the existence of solutions for second order abstract differential equations. For the theory of cosine functions of operators we refer to [29][30][31][32][33][34]. We next only mention a few concepts and properties relative to the second order abstract Cauchy problem.…”

Section: The Second Order Abstract Cauchy Problemmentioning

confidence: 99%

Second Order Impulsive Retarded Differential Inclusions with Nonlocal Conditions

Henrı́quez

Rabelo

Vale

2014

Abstract and Applied Analysis

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show abstract

“…In view of Proposition C.7.2 of [20], both −( L + κ I ) and L + κ I are m -accretive and {Reλ ≠ 0} ⊆ ρ( L + κ I ). Hence we have that

‖ {(λ^{2} + c λ - L)}^{- 1} ‖ \leq \frac{M}{| Re (λ^{2} + c λ + κ) |}

for some constant M > 0 as long as Re(λ 2 + c λ + κ) ≠ 0.…”

Section: Uniqueness and Monotonicity Of Periodic Traveling Wave Somentioning

confidence: 99%

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

Zhao

Ruan

2011

Journal de Mathématiques Pures et Appliquées

104

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We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka-Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c* such that for each wave speed c ≤ c*, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c < c* are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c > c*.

show abstract

The Functional Calculus for Sectorial Operators

Cited by 422 publications

References 128 publications

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

Second Order Impulsive Retarded Differential Inclusions with Nonlocal Conditions

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

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