In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

The Functional Calculus for Sectorial Operators

A definition of fractional (or complex) powers A°>, a e C, is given for closed linear operators A in a Banach space X with the resolvent set containing the negative real ray (—oo, 0) and such that {λ(λ + A)\"; 0 < λ < oo} is bounded; fundamental properties such as additivity (A*A = AP), coincidence with the iterations A* = A for integers a = n, and analytic dependence on a are discussed. Since the fractional powers A* are generally unbounded in both of the cases Re a > 0 and Re a < 0, attention is paid to the domains D{A), which are related to the spaces D and R of xeX defined by the regularity of (Λ + A)\"^ at oo and 0. When —A generates a bounded continuous semi-group or a bounded analytic semi-group, more detailed results are obtained.

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Fractional powers of operators.

Numerical differentiation in noisy environment is revised through an algebraic approach. For each given order, an explicit formula yielding a pointwise derivative estimation is derived, using elementary differential algebraic operations. These expressions are composed of iterated integrals of the noisy observation signal. We show in particular that the introduction of delayed estimates affords significant improvement. An implementation in terms of a classical finite impulse response (FIR) digital filter is given. Several simulation results are presented.

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Numerical differentiation with annihilators in noisy environment

/pdf/fractional-powers-of-dissipative-operators-1qwy7aytgt.pdf

Fractional powers of dissipative operators

: This analysis concerns the study of equations of the form Lu = 1 dui dt - Au = 0 for functions u(t) with values in some Banach space, as well as inhomogeneous equations Lu = f and ligh ly perturbed equations, with the main emphasis on the behavior of solutions a t + . Treated are equations for which the initial value probl (prescribing u at some value of t) is not necessarily well posed. Particularly consi ered are equations arising from partial differential equations in a cylinder (with t axis along the g nerator) which may be elliptic, and for which, therefore the initial value problem is indeed not well posed. The operator A then represents a partial differential operator in the variables in the base of the cylinder. ( ut or)

Properties of Solutions of Ordinary Differential Equations in Banach Space

Lectures on Differential and Integral Equations.

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Fractional Powers of Infinitesimal Generators and the Analyticity of the Semi-groups Generated by Them

Operateur d'integration h et operateur de differentiation S (classes des hyperfonctions: C et C H ). Equations differentielles ordinaires lineaires a coefficients lineaires (la classe C/C des hyperfonctions). Operateur de decalage exp(−λS) et operateur de diffusion exp(−λS 1/2 ). Applications aux equations aux derivees partielles

Operational Calculus: A Theory of Hyperfunctions

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On the differentiability of semi-groups of linear operators

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1261157656

Kôsaku Yosida

Papers

Lectures on Differential and Integral Equations.

Fractional Powers of Infinitesimal Generators and the Analyticity of the Semi-groups Generated by Them

Operational Calculus: A Theory of Hyperfunctions

On the differentiability of semi-groups of linear operators

An operator-theoretical integration of the wave equation.