On some properties of the multidimensional Bochner-Phillips functional calculus

Миротин, А. Р.

doi:10.1134/s0037446611060085

Cited by 16 publications

(20 citation statements)

References 11 publications

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“…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: 99%

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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Section: Main Results: Holomorphy and Preservation Of Anglesmentioning

confidence: 99%

On subordination of holomorphic semigroups

Gomilko

Tomilov

2015

Advances in Mathematics

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“…Denote by

i X_{p^{'}}

and

L_{p^{'}}

the operators

i X

and L when considered with their respective domains on

L^{p^{'}} false(G false)

. Then, from [, Lemma 11] we have

σ_{J} false(L, - i X false) \supseteq σ_{R} false(L, - i X false) = σ_{a} false({false(i X false)}^{*}, L^{*} false) = σ_{a} (i X_{p^{'}}, L_{p^{'}}) \supseteq E_{p^{'}} = {normalΔ}^{p} false(L, - i X false) .

The proof of

{normalΔ}^{p} false(L, - i X false) \subseteq σ_{J} false(L, - i X false)

is thus completed, hence, also the proof of Theorem .

□

…”

Section: Joint Spectral Multipliers Of (L−ix)mentioning

confidence: 99%

Spectral multipliers for functions of fixed K‐type on Lp(SL(2,R))

Ricci

Wróbel

2020

Mathematische Nachrichten

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“…As was mentioned in [17], the one-dimensional Bochner-Phillips functional calculus is a substantial part of the theory of operator semigroups and finds important applications in the theory of random processes (see, e.g., [19], [20]). The foundations of multidimensional calculus were laid by the author in [21], [22], [23], [24], [25]. Below we recall some notions and facts from [6], [21], [22], and [23], which we need for formulating our results.…”

Section: Introductionmentioning

confidence: 99%

“…In the one-dimensional case, it is called the semigroup subordinate to T A . In [27] it was noticed that the closure of the operator ψ(A) exists and is the generator of the C 0 semigroup g(A) (cf. [23].)…”

Section: Introductionmentioning

confidence: 99%

“…Then∆ ′ B/A (z) ∆ B/A (z) = tr(R(z, B) − R(z, A)), z ∈ ρ(A) ∩ ρ(B).Proof. Differentiating(24) we get for z ∈ ρ(A) ∩ ρ(B) in view of corollary 4∆ ′ B/A (z) ∆ B/A (z) = ξ A,B (t), d dz 1 t + z = − ξ ′ A,B (t), 1 t + z = − η A,B (t), 1 t + z = tr(R(z, B) − R(z, A)).Formula (20) implies that lim λ→+∞ ∆ B/A (λ) = 1. It follows also from the definition 6 and corollary 8 that ∆ B/A (z)∆ C/B (z) = ∆ C/A (z) for z ∈ C \ (−∞, 0], and operators A, B, C ∈ Gen(X) such that the pairs (A, B) and (B, C) satisfy all the conditions of corollary 6.…”

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confidence: 99%

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Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, II

Миротин¹

2018

Operators and Matrices

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The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreȋn trace formula derived. The aim of the present paper is to prove the Frechet differentiability of operator Bernstein functions and the Livschits-Kreȋn trace formula in the multidimensional setting.

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On some properties of the multidimensional Bochner-Phillips functional calculus

Cited by 16 publications

References 11 publications

On subordination of holomorphic semigroups

On subordination of holomorphic semigroups

Spectral multipliers for functions of fixed K‐type on Lp(SL(2,R))

Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, II

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