On the τ-calculus of generators forC 0-semigroups

Миротин, А. Р.

doi:10.1007/bf02673905

Cited by 14 publications

(29 citation statements)

References 3 publications

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“…The present paper is devoted to some generalizations and analogs of Yosida's Theorem in terms of so-called Bochner-Phillips calculus [1,14] (see also [5,Chap. XIII]; [8,15,11,2]). Though the majority of works on Bochner-Phillips calculus use the class B of (positive) Bernstein functions, we prefer the class T of negative one.…”

Section: Introductionmentioning

confidence: 99%

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Criteria for analyticity of subordinate semigroups

Миротин

2008

Semigroup Forum

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Let ψ be a Bernstein function. A. Carasso and T. Kato obtained necessary and sufficient conditions for ψ to have a property that ψ(A) generates a quasibounded holomorphic semigroup for every generator A of a bounded C 0 -semigroup in a Banach space, in terms of some convolution semigroup of measures associated with ψ. We give an alternative to Carasso-Kato's criterium, and derive several sufficient conditions for ψ to have the above-mentioned property.2000 Mathematics Subject Classification: 47A60, 47D03

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Section: Introductionmentioning

confidence: 99%

“…The closure of this operator, which is also denoted by ψ(A), is a generator of a bounded C 0semigroup g t (A) on X (the "subordinate semigroup"), too. (For the multidimensional version of this calculus see, e.g., [9], [10], [11]. )…”

Section: Introductionmentioning

confidence: 99%

Criteria for analyticity of subordinate semigroups

Миротин

2008

Semigroup Forum

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“…Since the choice of θ ∈ (0, θ) is arbitrary, the corollary follows. 2 Unfortunately, the proof of this fact in [21] seems to contain a mistake. Specifically, in the notation of [21], the proof at its final stage relies on the boundedness of the operator ψ(A)g t (A) which was not proved in [21].…”

Section: Main Results: Holomorphy and Preservation Of Anglesmentioning

confidence: 92%

“…2 Unfortunately, the proof of this fact in [21] seems to contain a mistake. Specifically, in the notation of [21], the proof at its final stage relies on the boundedness of the operator ψ(A)g t (A) which was not proved in [21]. Nonetheless, the holomorphy of (e −tψ(A) ) t≥0 was proved in [24,Theorem 13] for uniformly convex X by means of the Kato-Pazy criterion.…”

Section: Main Results: Holomorphy and Preservation Of Anglesmentioning

confidence: 92%

On subordination of holomorphic semigroups

Gomilko

Tomilov

2015

Advances in Mathematics

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We prove that for any Bernstein function ψ the operator −ψ(A) generates a bounded holomorphic C 0 -semigroup (e −tψ(A) ) t≥0 on a Banach space, whenever −A does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that (e −tψ(A) ) t≥0 is holomorphic in the holomorphy sector of (e −tA ) t≥0 , and if (e −tA ) t≥0 is sectorially bounded in this sector then (e −tψ(A) ) t≥0 has the same property. We also obtain new sufficient conditions on ψ in order that, for every Banach space X, the semigroup (e −tψ(A) ) t≥0 on X is holomorphic whenever (e −tA ) t≥0 is a bounded C 0 -semigroup on X. These conditions improve and generalize well-known results by Carasso-Kato and Fujita.

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“…8, §2] and also [8][9][10]) and natural applications related to solvability issues for the Cauchy problem for abstract first-order differential equations unsolved for the derivative. (See [11] and references therein.)…”

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confidence: 99%

On the inverse of the generator of a bounded C 0-semigroup

Gomilko

2004

Funct Anal Its Appl

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Let A be the generator of a uniformly bounded C 0 -semigroup in a Banach space B , and let A have a densely defined inverse A −1 . We present sufficient conditions on the resolvent (A−λI) −1 , Re λ > 0, under which A −1 is also the generator of a uniformly bounded C 0 -semigroup.Key words: uniformly bounded C 0 -semigroup, inverse of the generator, Banach space, Carleson embedding theorem. 1.Let B be a Banach space with norm · , and let E = E(B) be the set of densely defined closed linear operators in B. We denote the domain of an operator A ∈ E by D(A), the spectrum of A by σ(A), the identity operator by I , and the resolvent ofthe set of operators whose spectra lie in the half-plane Re λ 0. If x ∈ B and y ∈ B * , where B * is the dual Banach space of B with norm · * , then (x, y) stands for the value of the functional y on x.Let us recall some facts of operator semigroup theory. (See [1, Chap. 9] and [2,In what follows, an operator C 0 -semigroup is called simply a semigroup for brevity. The generator A ∈ E of T (t) is defined by the formulaand we write T (t) = e tA , t 0. A semigroup T (t) is said to be uniformly bounded if there exists a constant M 1 such that T (t) M , t 0. By G = G (B) we denote the set of generators of uniformly bounded semigroups in B.Let H (θ) [1, Chap. 9, §1] be the set of generators of semigroups holomorphic (analytic) in the sector Σ θ = {z = re iφ : 0 < r < ∞, |φ| < θ}, θ ∈ (0, π/2], and uniformly bounded in an arbitrary sector Σ θ 0 , θ 0 ∈ (0, θ). Next, let H 0 be the set of operators of the form A = A 0 + βI , where β 0 and A 0 ∈ H (θ) for some θ ∈ (0, π/2]. The set H 0 consists of generators of semigroups holomorphic in some sector Σ θ and, for β > 0, quasibounded in the terminology in [1, Chap. 9, §1.4].If A ∈ H (θ) has an inverse A −1 ∈ E (i.e., ker A = {0} and the range of A is dense in B), then A −1 also belongs to H (θ) [3,4]; in particular, A −1 ∈ G . Indeed [1, 2], A ∈ H (θ) if and only if the resolvent R(A, λ) exists for λ ∈ Σ π/2+θ and satisfies the estimateTherefore, if A ∈ H (θ) and A −1 ∈ E , thenfor λ ∈ Σ π/2+θ , and relations (1) and (2) readily imply the estimate R(A −1 , λ) (1 + c φ )|λ| −1 , λ ∈ Σ π/2+φ , for all φ ∈ (0, θ). We see that A −1 ∈ H (θ).In this paper, we extend the implication

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On the τ-calculus of generators forC 0-semigroups

Cited by 14 publications

References 3 publications

Criteria for analyticity of subordinate semigroups

Criteria for analyticity of subordinate semigroups

On subordination of holomorphic semigroups

On the inverse of the generator of a bounded C 0-semigroup

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