Eigenvalue estimates for the resolvent of a non-normal matrix

Reeb

Wolf

2014

Commun. Math. Phys.

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We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. We use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map. Finally, we discuss how general detailed balance conditions for quantum Markov processes lead to spectral convergence bounds. Contents

Section: B Spectral Bounds For the Convergence Of Markovian Processementioning

confidence: 73%

Spectral Convergence Bounds for Classical and Quantum Markov Processes

Reeb

Wolf

2014

Commun. Math. Phys.

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“…Section 2.3), the original estimate (4) in fact is stronger than the inequality in (2) and it is possible to prove (1) with C n = 2 2− 1 n starting from (4). To this end we bound the resolvent using advanced methods from the theory of model operators [15,23]. These techniques naturally lead us to estimate a hyperbolic analogue of the optimal matching distance and to the more sophisticated interpolation with finite Blaschke products.…”

Section: Classical Methods For Spectral Variation Boundsmentioning

confidence: 99%

“…On the technical side, our article contains two key innovations to the methods developed in the cited publications. First, we employ recent spectral resolvent estimates [23,15] that are stronger than the Hadamard-type inequality (2) used by Bhatia, Elsner and Krause. These resolvent estimates are derived using an interpolation-theoretic approach to eigenvalue bounds introduced in [15].…”

Section: Introductionmentioning

confidence: 99%

Spectral variation bounds in hyperbolic geometry

Müller-Hermes

Linear Algebra and its Applications

2015

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We derive new estimates for distances between optimal matchings of eigenvalues of non-normal matrices in terms of the norm of their difference. We introduce and estimate a hyperbolic metric analogue of the classical spectral-variation distance. The result yields a qualitatively new and simple characterization of the localization of eigenvalues. Our bound improves on the best classical spectralvariation bounds due to Krause if the distance of matrices is sufficiently small and is sharp for asymptotically large matrices. Our approach is based on the theory of model operators, which provides us with strong resolvent estimates. The latter naturally lead to a Chebychev-type interpolation problem with finite Blaschke products, which can be solved explicitly and gives stronger bounds than the classical Chebychev interpolation with polynomials. As compared to the classical approach our method does not rely on Hadamard's inequality and immediately generalizes to algebraic operators on Hilbert space.

“…We first prove (9). We consider T ∈ C n with spectrum σ and assume for a moment that σ 2 ⊂ D. It follows from [OS,Theorem III.2] that for any ζ ∈ C − σ the resolvent of T is bounded by…”

Section: Proofsmentioning

confidence: 99%

“…It is well known [NN1,Theorem 3.12,p.147], [OS,Lemma III.5] that (see the cited articles for a discussion)…”

Section: Proofsmentioning

confidence: 99%

Maximum of the resolvent over matrices with given spectrum

Journal of Functional Analysis

Zarouf

2017

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In numerical analysis it is often necessary to estimate the condition number CN (T ) = ||T | | · T −1 and the norm of the resolvent (ζ − T ) −1 of a given n × n matrix T . We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the well-known fact that the supremum of CN (T ) over all matrices with ||T | | ≤ 1 and minimal absolute eigenvalue r = min i=1,...,n |λ i | > 0 is the Kronecker bound 1 r n . This result is subsequently generalized by computing the corresponding supremum of (ζ − T ) −1 for any |ζ| ≤ 1. We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occuring Toeplitz matrices are socalled model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space H 2 to a finite-dimensional invariant subspace.