Explicit upper bounds for the spectral distance of two trace class operators

Bandtlow, Oscar F.; Guven, Ayse

doi:10.1016/j.laa.2014.10.008

Cited by 3 publications

(1 citation statement)

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“…In [35], it is shown that in general, there is no convergence of spectra and it is determined, for which subsets K of C there exists a filtration {P n } such that d H (σ(T n ), K) → 0 as n → ∞, where d H denotes the Hausdorff distance. On the other hand, there are also some positive results assuring the convergence of σ(T n ) to σ(T ) under some restrictive hypotheses, see [24,6] and references in [6]. Proposition 4.2 in [10] contains an abstract result on the partial limit set of ε-pseudospectra of T n , under certain hypotheses.…”

Section: General Theorems On Convergencementioning

confidence: 99%

Infinite-dimensional features of matrices and pseudospectra

Pal

Yakubovich

2017

Journal of Mathematical Analysis and Applications

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Abstract. Given a Hilbert space operator T , the level sets of function Ψ T (z) = (T − z) −1 −1 determine the so-called pseudospectra of T . We set Ψ T to be zero on the spectrum of T . After giving some elementary properties of Ψ T (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator T , there is a sequence {T n } of finite matrices such that Ψ Tn (z) tends to Ψ T (z) uniformly on C. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices.One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen-Douglas class operators. We use these results to show that, to the opposite to the function Ψ S , the function √ S − z for certain finite matrices S may oscillate arbitrarily fast even far away from the spectrum.

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