About stability estimates and resolvent conditions
TL;DRAbstract
(1.1) ‖B‖ ≤M0 for n = 0, 1, 2, . . . , where M0 is a positive constant. For the time being ‖ · ‖ stands for the spectral norm (i.e. the matrix norm induced by the Euclidean norm on C). The famous Kreiss matrix theorem (see e.g. [11], [32]) relates (1.1) to conditions on B which are easier to verify than (1.1). One of these conditions involves the so-called resolvent (ζI −B)−1 of B, and reads as follows: (1.2) ζI − B is invertible and ‖(ζI − B)−1‖ ≤ M1(|ζ| − 1)−1 for all complex numbers ζ 6∈ D. Here M1 is a positive constant, I the s × s identity matrix and D = {ζ : ζ ∈ C and |ζ| ≤ 1} the closed unit disk.
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(1.1) ‖B‖ ≤M0 for n = 0, 1, 2, . . . , where M0 is a positive constant. For the time being ‖ · ‖ stands for the spectral norm (i.e. the matrix norm induced by the Euclidean norm on C). The famous Kreiss matrix theorem (see e.g. [11], [32]) relates (1.1) to conditions on B which are easier to verify than (1.1). One of these conditions involves the so-called resolvent (ζI −B)−1 of B, and reads as follows: (1.2) ζI − B is invertible and ‖(ζI − B)−1‖ ≤ M1(|ζ| − 1)−1 for all complex numbers ζ 6∈ D. Here M1 is a positive constant, I the s × s identity matrix and D = {ζ : ζ ∈ C and |ζ| ≤ 1} the closed unit disk.
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