Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

Hagen Neidhardt (), Artur Stephan () and Valentin A. Zagrebnov ()
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Hagen Neidhardt: Weierstrass Institute for Applied Analysis and Stochastics
Artur Stephan: Weierstrass Institute for Applied Analysis and Stochastics
Valentin A. Zagrebnov: Institut de Mathématiques de Marseille (UMR 7373), Université d’Aix-Marseille, CMI - Technopôle Château-Gombert

A chapter in Analysis and Operator Theory, 2019, pp 271-299 from Springer

Abstract: Abstract The paper is devoted to evolution equations of the form $$\begin{aligned} \frac{\partial }{\partial t}u(t) = -(A + B(t))u(t), \quad t \in {\mathcal {I}}= [0,T], \end{aligned}$$ ∂ ∂ t u ( t ) = - ( A + B ( t ) ) u ( t ) , t ∈ I = [ 0 , T ] , on separable Hilbert spaces where A is a non-negative self-adjoint operator and $$B(\cdot )$$ B ( · ) is family of non-negative self-adjoint operators such that $$\mathrm {dom}(A^{\alpha }) \subseteq \mathrm {dom}(B(t))$$ dom ( A α ) ⊆ dom ( B ( t ) ) for some $${\alpha }\in [0,1)$$ α ∈ [ 0 , 1 ) and the map $$A^{-{\alpha }}B(\cdot )A^{-{\alpha }}$$ A - α B ( · ) A - α is Hölder continuous with the Hölder exponent $${\beta }\in (0,1)$$ β ∈ ( 0 , 1 ) . It is shown that the solution operator U(t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by A and B(t) provided the condition $${\beta }> 2{\alpha }-1$$ β > 2 α - 1 is satisfied. The convergence rate for the approximation is given by the Hölder exponent $${\beta }$$ β . The result is proved using the evolution semigroup approach.

Date: 2019
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DOI: 10.1007/978-3-030-12661-2_13

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