A stability result for sectorial operators in branch spaces

In this paper the author considers approximations of holomorphic semigroups on general Banach spaces based on A-acceptable rational functions. A general stability theorem is proved. It covers rational functions $r(z)$ with $|r(\infty )| = 1$ and can be applied even to variable step cases. In particular the question of the ${\text{L}}_p $-stability $(p \ne 2)$ of Crank-Nicolson schemes for parabolic initial value problems is settled.

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