Journal of Spectral Theory

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Volume 2, Issue 4, 2012, pp. 355–371
DOI: 10.4171/JST/33

Polynomials of almost normal arguments in $C^*$-algebras

Nikolai D. Filonov[1] and Ilya Kachkovskiy[2]

(1) St. Petersburg Department of, Steklov Mathematical Institute, Fontanka 27, 191023, St. Petersburg, Russian Federation
(2) Department of Mathematics, King's College, Strand, WC2R 2LS, London, UK

The functional calculus for normal elements in $C^*$-algebras is an important tool of analysis. We suggest an approximate substitute for such calculus for elements $a$ with the small self-commutator norm $\|[a,a^*]\|\le \delta$. We show that many properties of the functional calculus are conserved up to order $\delta$.

Keywords: $C^*$-algebras, functional calculus, self-commutator, polynomials, Positivstellensatz, pseudospectrum

Filonov Nikolai, Kachkovskiy Ilya: Polynomials of almost normal arguments in $C^*$-algebras. J. Spectr. Theory 2 (2012), 355-371. doi: 10.4171/JST/33