scholarly journals Remarks on the Operator-Norm Convergence of the Trotter Product Formula

2018 ◽  
Vol 90 (2) ◽  
Author(s):  
Hagen Neidhardt ◽  
Artur Stephan ◽  
Valentin A. Zagrebnov
Keyword(s):  
Product Formula ◽  
Operator Norm ◽  
Norm Convergence ◽  
Trotter Product Formula

TROTTER PRODUCT FORMULA FOR NONSELF-ADJOINT GIBBS SEMIGROUPS

2001 ◽  
Vol 64 (2) ◽  
pp. 436-444 ◽  
Author(s):  
VINCENT CACHIA ◽  
VALENTIN A. ZAGREBNOV
Keyword(s):  
Convergence Rate ◽  
Error Bound ◽  
Trace Class ◽  
Product Formula ◽  
Trace Norm ◽  
Norm Convergence ◽  
Trotter Product Formula

The trace-norm convergence of the Trotter product formula is proved for nonself-adjoint Gibbs semigroups. For any m-sectorial generators A and B such that e−tReA is in the trace-class for t > 0, the Trotter product formula converges in the trace-norm. With smallness conditions on B with respect to A, we give error bound estimates of the convergence rate in this topology.

scholarly journals The norm estimate of the difference between the Kac operator and the Schrödinger semigroup: A unified approach to the nonrelativistic and relativistic cases

1998 ◽  
Vol 149 ◽  
pp. 53-81 ◽  
Author(s):  
Takashi Ichinose ◽  
Satoshi Takanobu
Keyword(s):  
Schrödinger Operators ◽  
Product Formula ◽  
Operator Norm ◽  
Schrodinger Operators ◽  
Unified Approach ◽  
Trotter Product Formula ◽  
Nonrelativistic Case ◽  
Norm Estimate ◽  
Schrödinger Semigroup ◽  
The Difference

Abstract.An Lp operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup is proved and used to give a variant of the Trotter product formula for Schrödinger operators in the Lp operator norm. The method of the proof is probabilistic based on the Feynman-Kac formula. The problem is discussed in the relativistic as well as nonrelativistic case.

Trotter–Kato Product Formula and Operator-Norm Convergence

1999 ◽  
Vol 205 (1) ◽  
pp. 129-159 ◽  
Author(s):  
H. Neidhardt ◽  
V. A. Zagrebnov
Keyword(s):  
Product Formula ◽  
Operator Norm ◽  
Norm Convergence

scholarly journals Approximations of self-adjoint \(C_0\)-semigroups in the operator-norm topology

2020 ◽  
Vol 73 (2) ◽  
pp. 175
Author(s):  
Valentin Zagrebnov
Keyword(s):  
Approximation Theory ◽  
Product Formula ◽  
Operator Norm ◽  
Optimal Estimate ◽  
Norm Convergence ◽  
Norm Topology

The paper improves approximation theory based on the Trotter–Kato product formulae. For self-adjoint \(C_0\)-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or product) formula to convergence in the operator-norm topology. This allows to obtain optimal estimate for the rate of operator-norm convergence of Trotter–Kato product formulae for Kato functions from the class \(K_2\).

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