Please use this identifier to cite or link to this item:
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
ISSN: 08954798
Appears in Collections:Staff Publications

Show full item record
Files in This Item:
There are no files associated with this item.

Page view(s)

checked on Nov 9, 2018

Google ScholarTM


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.