Linear Equations in Banach Spaces

Крейн, С. Г.; Iacob, Alina

doi:10.1007/978-1-4684-8068-9

Cited by 158 publications

(116 citation statements)

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“…Clearly, by Plancherel's theorem, (2.9) is also necessary. Adapting Krein's construction from [38], see also [36,Example 3.4], one can show that the rate of decay of the L 2 norms in (2.9) (with α = 1 2 ) cannot be weakened, in general.…”

Section: Recall That a Banach Space X Is Called A Umd Space If The Himentioning

confidence: 99%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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Abstract. The paper provides new characterisations of generators of cosine functions and C 0 -groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0 -groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini's result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh's characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0 -semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.

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Section: Recall That a Banach Space X Is Called A Umd Space If The Himentioning

confidence: 99%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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“…Starting with k = 1, we can successively find from (13) all u m k (t k ). The operator coefficients U m (t, s) can concurrently be found in advance.…”

Section: U (T S)f (S) Dsmentioning

confidence: 99%

A Parallel High-accuracy Method for the First-order Evolution Equation in Hilbert and Banach Spaces

Garvrilyuk¹,

Макаров

Ryabichev

2003

Comput. Methods Appl. Math.

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-We have developed a discretization method of an arbitrarily given order of accuracy with respect to the time discretization parameter for the first-order evolution differential equation in Banach space. The method includes two levels of parallelism: the operator exponential needed for the calculation of the evolution operator can be computed in parallel analogously to [6] (inner parallelism) and then we can compute in parallel the evolution operator at various points of the time mesh (outer parallelism).2000 Mathematics Subject Classification: 65F50; 65F30; 15A09; 15A24; 15A99.

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“…It is obvious that M h (α) is an analytic family related to α in Kato's sense (see [6, p. 24]), and hence it is possible to use the Kato-Rellix's Theorem from the regular theory of perturbations (see Theorem 12-13 of [6]). Combining all results above mentioned, we obtain the following theorem: …”

Section: Let Us Represent the Operator [Mmentioning

confidence: 99%

“…Let us find the coefficients a k (α) and b k (α) in the expression of U k , which depend on the initial conditions (5) and (6).…”

Section: π) Let Us Denote This Solution By (U G)mentioning

confidence: 99%

Guided waves in a fluid layer on an elastic irregular bottom

Collar¹

1996

Publicacions Matemàtiques

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In this paper one considers the linearized problem to determine the movement of an ideal heavy fluid contained in an unbounded container with elastic walls. As initial data one knows the movement of both the bottom and the free surface of the fluid and also the strength of certain perturbation, strong enough to take the bottom out of its rest state.One important point to be considered regards the influence of the bottom's geometry on the propagation of superficial waves. This problem has been already studied in other works without considering the elastic properties of the bottom and considering a cilindrical container with bounded section.

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Linear Equations in Banach Spaces

Cited by 158 publications

References 0 publications

Resolvent characterisation of generators of cosine functions and C 0-groups

Resolvent characterisation of generators of cosine functions and C 0-groups

A Parallel High-accuracy Method for the First-order Evolution Equation in Hilbert and Banach Spaces

Guided waves in a fluid layer on an elastic irregular bottom

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