Extensions of the multicentric functional calculus

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School of Science | Doctoral thesis (article-based) | Defence date: 2021-12-10
Degree programme
42 + app. 88
Aalto University publication series DOCTORAL DISSERTATIONS, 174/2021
In operator theory, one of the central concepts is the spectrum of an operator and if one knows how to separate the spectrum into components, then the multicentric calculus is a useful tool, introduced by Olavi Nevanlinna in 2011. This thesis presents extensions of the multicentric calculus from single operators to n-tuples of commuting operators, for both holomorphic and non-holomorphic functions. It also covers the same calculus when replacing the polynomials with rational functions. The multicentric representation of holomorphic functions gives a simple way to generalize the von Neumann result, i.e., the unit disc is a spectral set for contractions in Hilbert spaces. In other words, this calculus provides a way of representing the spectrum of a bounded operator T, by searching for a polynomial p that maps the spectrum to a small disc around origin. Since the von Neumann inequality works for contractions with spectrum in the unit disc, the multicentric representation applies a suitable polynomial p to the operator T, so that p(T) becomes a contraction with spectrum in the unit disc and thus the usual holomorphic functional calculus holds. When extending the calculus to n-tuples of commuting operators, a constant and some extra conditions are needed for the von Neumann inequality to hold true. In order to extend the calculus to non-holomorphic functions, the Banach algebra is the tool to use in finding those functions for which it is possible to have a simple functional calculus by using suitable polynomial p. For a given bounded operator T on a Hilbert space, the polynomial p is such that p(T) is diagonalizable or similar to normal. The operators here are considered to be matrices. In particular, the calculus provides a natural approach to deal with non-trivial Jordan blocks. The extension of this calculus is done for a pair of commuting matrices, by constructing a suitable tensor product Banach algebra that can be identified with the space of continuous functions of two variables. When replacing the polynomial by a rational function, one can apply the calculus to functions that are not polynomially convex, thus extending it to a larger class of functions.
Supervising professor
Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
Thesis advisor
Nevanlinna, Olavi, Prof., Emeriti, Aalto University, Finland
multicentric calculus, lemniscates, spectral projection, commuting operators, Banach algebra, tensor norm, rational functions, series expansions
Other note
  • [Publication 1]: Diana Apetrei, Olavi Nevanlinna. Multicentric calculus and the Riesz projection. Journal of Numerical Analysis and Approximation Theory, Volume 44, no.2, pp. 127-145, 2015
  • [Publication 2]: Diana Andrei. Multicentric holomorphic calculus for n−tuples of commuting operators. Advances in Operator Theory, Volume 4, no.2, pp. 447-461, 2019.
    DOI: 10.15352/aot.1804-1346 View at publisher
  • [Publication 3]: Diana Andrei. A tensor product algebra with functional calculus for commuting pair of matrices. Submitted to Advances in Operator Theory, September 22th 2021
  • [Publication 4]: Diana Andrei, Olavi Nevanlinna, Tiina Vesanen. Rational functions as new variables. Submitted to Banach Journal of Mathematical Analysis, May 24th 2021