The Application of C-Semigroups to Differential Operators in Lp(Rn)

Lei, Yansong; Zheng, Qian

doi:10.1006/jmaa.1994.1464

Cited by 8 publications

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“…The existence of these regularized semigroups and cosine functions is known (see [10], [15], [16], [4, Chapter XIII], [3], [12], [13]); we offer our approach as a simple, intuitive, constructive and unified corollary of our regularized functional calculus. For example, on L p (R n )(1 ≤ p < ∞), we may simultaneously deal with the Schrödinger equation (ill-posed for p = 2) and the wave equation (illposed for p = 2, n > 1), by constructing a regularized BC k ((−∞, 0]) functional calculus for the Laplacian.…”

Section: Introductionmentioning

confidence: 99%

Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

deLaubenfels

Lei

1997

Abstract and Applied Analysis

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Abstract. Let iA j (1 ≤ j ≤ n) be generators of commuting bounded strongly continuous groups, A ≡ (A 1 , A 2 , ..., A n ). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k, r > 0 such that f (A) has a (1+|A| 2 ) −r -regularized BC k (f (R n )) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when IntroductionIn finite dimensions, the Jordan canonical form for matrices guarantees that, although a linear operator may not be diagonalizable, which is equivalent to having a BC(C) functional calculus, it will be generalized scalar, that is, have a BC k (C) functional calculus, for some k; specifically, k may be chosen to be n − 1, where n is the order of the largest Jordan block.In infinite dimensions, even a bounded linear operator on a Hilbert space may fail to be generalized scalar; consider the left shift on 2 .Our favorite unbounded operators fail to be generalized scalar, on Banach spaces that are not Hilbert spaces. The operator i d dx , on L 2 (R), is selfadjoint and thus has a BC(R) functional calculus. However, on L p (R), p = 2, it does not have a BC m (R) functional calculus, for any nonnegative integer m; that is, it is not even generalized scalar (see [2, Lemma 5.3]).

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Section: Introductionmentioning

confidence: 99%

Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

deLaubenfels

Lei

1997

Abstract and Applied Analysis

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“…As canonical examples, we consider in Section 4 a class of differential operators on certain function spaces L p l (R n ), (1< p< , l=0, ..., n ; L p 0 (R n ) is just the usual Banach space L p (R n )), and show that they generate integrated or regularized semigroups. In the case of L p (R n ), our results (Theorems 4.4 and 4.5) perfect the corresponding results in [12,18]. Finally, Section 5 presents some applications to Cauchy problems.…”

Section: Introductionmentioning

confidence: 63%

“…Specialized to the Banach space L p (R n ) (1<p< ), Theorem 4.5 perfects Theorem 2.3 in [18], by allowing r to take the critical value n | 1 2 &(1Âp)|. It also takes care of a few gaps in the proof there.…”

Section: Differential Operators As Generatorsmentioning

confidence: 96%

Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces

Xiao

Liang

1997

Journal of Functional Analysis

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In this paper, we consider the vector-valued Laplace transforms, r-times (r # [0, )) integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces. Our theorems develop the corresponding results in [1,11], including the well known integrated version of the classical Widder's representation theorem of Laplace transforms for functions taking values in Banach spaces. Moreover, we study a class of differential operators on certain function spaces. Optimal conditions, making them the generators of integrated or regularized semigroups, are obtained. We finally show some applications to abstract Cauchy problems.1997 Academic Press

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“…Lemma 2.5. [14] Let M ≥ 0, ω ∈ R and A be a linear operator on X. Then the following conditions are equivalent:…”

Section: The Generator Amentioning

confidence: 99%

Necessary and sufficient conditions for the approximate controllability of fractional linear systems via C−semigroups

Lian

Fan

2022

Filomat

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The approximate controllability of fractional linear evolution systems is considered in this paper. Firstly, the definitions of the mild solution and the approximate controllability of fractional linear evolution systems are obtained by using the theory of C?semigroups. Secondly, a new set of necessary and sufficient conditions are established to examine that linear system is approximately controllable with the help of symmetric operator. Moreover, the restricted condition of the state space is weakened by means of the dual mapping. Finally, as applications, the approximate controllability of nonlinear evolution systems are derived under the assumption that the corresponding linear system is approximately controllable. Our work essentially improves and generalized the corresponding results which are based on strongly continuous semigroups.

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The Application of C-Semigroups to Differential Operators in Lp(Rn)

Cited by 8 publications

References 0 publications

Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces

Necessary and sufficient conditions for the approximate controllability of fractional linear systems via C−semigroups

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