Landau-Kolmogorov Inequalities for Semigroups and Groups

Certain, Melinda W.; Kurtz, Thomas G.

doi:10.2307/2041794

Cited by 7 publications

(9 citation statements)

References 3 publications

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“…X , proving (2.2). (ii) A functional analytic proof of (2.2) was presented by Certain and Kurtz [13]. It uses Landau's inequality [75]…”

Section: Norm Inequalities For Generators Of C 0 Semigroupsmentioning

confidence: 99%

A survey of some norm inequalities

Gesztesy,

Nichols,

Stanfill

2021

Preprint

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We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type, and recall that under exceedingly stronger hypotheses on the operator A and/or the Banach space X , the optimal constant C in these inequalities diminishes from 4 (e.g., when A is the generator of a C 0 contraction semigroup on a Banach space X ) all the way down to 1 (e.g., when A is a symmetric operator on a Hilbert space H).We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.

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“…X , proving (2.2). (ii) A functional analytic proof of (2.2) was presented by Certain and Kurtz [13]. It uses Landau's inequality [75]…”

Section: Norm Inequalities For Generators Of C 0 Semigroupsmentioning

confidence: 99%

A survey of some norm inequalities

Gesztesy,

Nichols,

Stanfill

2021

Preprint

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“…The main result in this Appendix, Theorem B.2, is used in Part 5 of the proof of Theorem 3.3. It is an extension to generators of arbitrary C 0 semigroups of a well known estimate for generators of contraction semigroups (see [11]). Considering B−µI with a sufficiently large µ ≥ 0 instead of the original operator B, one can reduce the more general case to an estimate for the generator of a bounded semigroup (see (36) We start with the following Landau-Kolmogorov type inequality for n times continuously differentiable functions f : [0, ∞) → C:…”

Section: B Kolmogorov-kallman-rota Type Inequalitiesmentioning

confidence: 99%

“…Then (35) implies [11]). The optimal values of the constants M n,k are discussed in [32] and [11]. In particular, if n = 2, k = 1, then M n,k = M 2,1 = 4, (35) is the Landau [23] or [30, Chapter 1, Lemma 2.8]).…”

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confidence: 94%

Level Sets of the Resolvent Norm of a Linear Operator Revisited

Davies

Shargorodsky

2015

Mathematika

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It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space X cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space X and an unbounded, densely defined operator acting in X with a compact resolvent whose norm is constant in a neighbourhood of zero; moreover X is isometric to a Hilbert space on a subspace of co-dimension 2. There is also a bounded linear operator acting on the same space whose resolvent norm is constant in a neighbourhood of zero. It is shown that similar examples cannot exist in the co-dimension 1 case. * MSC2010 subject classification 47A10, 47D06 †

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“…In this paper, the methods of these authors are modified to prove the analogous result for other function spaces on R. For variants and applications of such results, see, for example, [4,9,11]. In particular, our results apply to amalgams of V and £9, as defined and studied in, for example, [2,3,5,6,7].…”

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confidence: 91%