Existence Families, Functional Calculi and Evolution Equations

deLaubenfels, Ralph

doi:10.1007/bfb0073401

Cited by 173 publications

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“…See [4] and the references therein, for basic material on regularized semigroups and their relationship to the abstract Cauchy problem. …”

Section: Regularized Functional Calculi Regularized Semigroups and Rmentioning

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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“…See [4] and the references therein, for basic material on regularized semigroups and their relationship to the abstract Cauchy problem. …”

Section: Regularized Functional Calculi Regularized Semigroups and Rmentioning

confidence: 99%

Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

deLaubenfels

Lei

1997

Abstract and Applied Analysis

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“…In general, the family (Y (t)) t≥0 is not even a C-existence family [8]. A deep analysis of this topic, that is, on the link of B-bounded semigroups with C-existence families and C-regularized semigroups can be found in [5].…”

Section: That a ∈ B-g(m ω X)mentioning

confidence: 99%

A new characterization of B‐bounded semigroups with application to implicit evolution equations

Arlotti¹

2000

Abstract and Applied Analysis

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We consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and called B-bounded semigroups. We first determine all the properties possessed by a couple (A, B) of operators if they generate a B-bounded semigroup (Y (t)) t≥0 . Then we determine the simplest further property of the couple (A, B) which can assure the existence of a C 0 -semigroup (T (t)) t≥0 such that for all t ≥ 0, f ∈ D(B) we can write Y (t)f = T (t)Bf . Furthermore, we compare our result with the previous ones and finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations.

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“…Since Davies and Pang ( [4]) rediscovered it in 1987, this semigroup has been extensively investigated (cf., e.g., [6,7,11,12,17,21,22]), because it can be used to deal with many ill-posed (in the classical sense) abstract Cauchy problems for which the strongly continuous semigroup is not applicable. In 1991, a new type of operator family, called the existence family, for controlling the first order abstract Cauchy problem, which is more general than the C-regularized semigroup, was introduced and discussed by deLaubenfels [5] (see also [6]). It proves to be more flexible in applications, because the existence family does not require commutativity among itself, its generator, and the regularizing operator (cf.…”

Section: Introductionmentioning

confidence: 99%

“…It proves to be more flexible in applications, because the existence family does not require commutativity among itself, its generator, and the regularizing operator (cf. [5,6]). The present paper is concerned with this type of operator family.…”

Section: Introductionmentioning

confidence: 99%