Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato

Hagen Neidhardt (1) , Artur Stephan (2) , Valentin A Zagrebnov (3)

1 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]
2 HU Berlin - Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin
3 I2M - Institut de Mathématiques de Marseille

The paper is devoted to evolution equations of the form ∂ ∂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(·) is family of non-negative self-adjoint operators such that dom(A α) ⊆ dom(B(t)) for some α ∈ [0, 1) and the map A −α B(·)A −α is Hölder continuous with the Hölder exponent 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 β > 2α − 1 is satisfied. The convergence rate for the approximation is given by the Hölder exponent β. The result is proved using the evolution semigroup approach.

Domaines: Analyse fonctionnelle [math.FA]

Hagen Neidhardt, Artur Stephan, Valentin A Zagrebnov. Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato. 2019. ⟨hal-01971610v1⟩