Please use this identifier to cite or link to this item: http://scholarbank.nus.edu.sg/handle/10635/104310
Title: The Kreiss matrix theorem on a general complex domain
Authors: Toh, K.-C. 
Trefethen, L.N.
Keywords: Conformal mapping
Faber polynomials
Kreiss matrix theorem
Krylov subspaces
Polynomials of a matrix
Issue Date: Aug-1999
Citation: Toh, K.-C.,Trefethen, L.N. (1999-08). The Kreiss matrix theorem on a general complex domain. SIAM Journal on Matrix Analysis and Applications 21 (1) : 145-165. ScholarBank@NUS Repository.
Abstract: Let A be a bounded linear operator in a Hilbert space H with spectrum A (A). The Kreiss matrix theorem gives bounds based on the resolvent norm ∥ (2I - A)-1 ∥ for ∥An ∥if A (A) is in the unit disk or for ∥etA∥ if A(A) is in the left half-plane. We generalize these results to a complex domain Ω, giving bounds for ∥ Fn (A) ∥ if A (A) ⊂ Ω, where Fn denotes the nth Faber polynomial associated with Ω. One of our bounds takes the form K̄ (Ω) ≤ 2 sup ∥ Fn (A) ∥, ∥ Fn (A) ∥ ≤ 2 e (n +) K̄(Ω), n where K̄(Ω) is the "Kreiss constant" defined by K̄(Ω) = inf { C : ∥ (zI - A)-1 ∥ ≤ C/dist(z, Ω) ∀ z ∉ Ω}. By means of an inequality due originally to Bernstein, the second inequality can be extended to general polynomials pn. In the case where H is finite-dimensional, say, dim(H) = N, analogous results are also established in which ∥ Fn (A)∥ is bounded in terms of N instead of n when the boundary of Ω is twice continuously differentiable.
Source Title: SIAM Journal on Matrix Analysis and Applications
URI: http://scholarbank.nus.edu.sg/handle/10635/104310
ISSN: 08954798
Appears in Collections:Staff Publications

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