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Spectral bounds using higher-order numerical ranges. (English) Zbl 1119.47004

Summary: This paper describes how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what are referred to here as ’its higher-order numerical ranges’. Proofs of some of their basic properties are given, as well as an explanation of how to compute them. Finally, they are used to obtain new spectral insights into the non-self-adjoint Anderson model in one and two space dimensions.

MSC:

47A10 Spectrum, resolvent
47B25 Linear symmetric and selfadjoint operators (unbounded)
47N55 Applications of operator theory in statistical physics (MSC2000)
60H25 Random operators and equations (aspects of stochastic analysis)
15A42 Inequalities involving eigenvalues and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

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