On the inverse Laplace transform for C 0 -semigroups in UMD-spaces

Driouich, A.; El-Mennaoui, O.

doi:10.1007/s000130050303

Cited by 11 publications

(14 citation statements)

References 8 publications

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“…This result brings out the intimate relationship between the control theory of infinite dimensional systems and the inverse Laplace transform. However, this characterization (Proposition 3.2) of L 2 -admissibility of control operators is useful, and yields a proof of the result by [9] on the validity of the (complex) inverse Laplace transform for C 0 -semigroups in UMD-Banach spaces. In the functional analysis area, by using our approach to the characterization of admissibility, we retrieve some well-known results on incomplete and complete inverse Laplace transforms of the resolvent of a semigroup in general Banach spaces and in UMD spaces, respectively.…”

Section: Introductionmentioning

confidence: 99%

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

Bounit

Driouich

El-Mennaoui

2010

Integr. Equ. Oper. Theory

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This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon-Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56-63, 1999) on the validity of the inverse formula of the Laplace transform for C0-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26 (5): [1331][1332][1333][1334][1335][1336][1337][1338][1339][1340][1341] 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00.

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Section: Introductionmentioning

confidence: 99%

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

Bounit

Driouich

El-Mennaoui

2010

Integr. Equ. Oper. Theory

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“…[14,Proposition 6]. Recall also that the complex inversion formula for C 0 -semigroups holds on every UMD space, see [14] or [2,Section 3.12]. In particular, it shows that there exists a generator A of a C 0 -(semi)group such that for any σ > 0 and p > 1 the following condition…”

Section: Recall That a Banach Space X Is Called A Umd Space If The Himentioning

confidence: 99%

“…As an example we can consider the shift group on X = L 1 (R), see e.g. [14,Proposition 6]. Recall also that the complex inversion formula for C 0 -semigroups holds on every UMD space, see [14] or [2,Section 3.12].…”

Section: Recall That a Banach Space X Is Called A Umd Space If The Himentioning

confidence: 99%

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Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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Abstract. The paper provides new characterisations of generators of cosine functions and C 0 -groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0 -groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini's result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh's characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0 -semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.

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“…From this one can then derive the standard result on semigroups (strong convergence on the domain of the generator). Driouich and El-Mennaoui [4] showed that in case that X has the c 2008 Australian Mathematical Society 1446-7887/08 $A2.00 + 0.00 74 M. Haase [2] unconditional martingale differences (UMD) property, the convergence is strong on all of X . This was subsequently generalized from semigroups to solution families for scalar-type Volterra equations by Cioranescu and Lizama in [3].…”

Section: Introductionmentioning

confidence: 99%