Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of a linear sectorial evolution equation ut = Au on a complex Banach space X. Given a strictly A(α)-stable q-step method of order p whose stability region includes a sectorial region containing the spectrum of the operator A, the corresponding evolution semigroup for the method is Cⁿ(hA), n ≥0, defined on X^q, where C(z) ∈ L (C^q) denotes the one-step map associated with the method. It is shown that for appropriately chosen V, Y: C → C^q, based on the principal right and left eigenvectors of C(z), Cⁿ(hA) approximates the semigroup V(hA)e^nhAY^H(hA) with optimal order p.