Operator Theory in Symmetrized Domains

Operator theory in symmetrized domains investigates how tuples of commuting operators on a Hilbert space can be analysed through the geometry of non-convex subsets of several complex variables. Central examples include the symmetrized bidisc, defined by the sum and product of two unit-disc variables, and its higher-dimensional analogues. Such domains arise naturally in control theory, where spectral sets govern system stability via μ-synthesis. The field unites dilation theory—seeking realisations of operators as compressions of normal tuples—with functional models developed through reproducing kernel Hilbert spaces. Core questions address when a given operator tuple admits a normal dilation whose spectrum lies in the distinguished boundary of a symmetrized domain, and how boundary interpolation problems can be solved by explicit analytic formulae. Recent advances have deepened our understanding of rational inner functions on symmetrized polydiscs, clarified the structure of spectral sets under automorphism groups, and connected complex‐geometric invariants to operator-theoretic inequalities. These developments not only enrich the abstract theory of multivariable operator models but also inform practical applications in robust control and signal processing.

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Operator Theory in Symmetrized Domains publication trend

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Technical terms

Symmetrized bidisc: The subset of C² given by {(z + w, zw) : |z|<1, |w|<1}, carrying non-convex complex geometry relevant to operator tuples.

Spectral set: A domain D in the complex plane (or Cⁿ) for which the spectrum of an operator (or operator tuple) lies in D and a von Neumann-type inequality holds for analytic functions on D.

Γ-contraction: A commuting pair of operators (S,P) for which the symmetrized bidisc acts as a spectral set, generalising single-variable contractions.

Rational dilation: A representation of an operator or tuple as the compression of a normal operator whose spectrum lies in the boundary of a given domain, realised via rational functional calculus.

Reproducing kernel: A function K(x,y) defining an inner product on a space of analytic functions so that point evaluation at y is given by ⟨f,K(·,y)⟩, enabling model-theoretic constructions.

References

1. Algebraic and geometric aspects of rational Γ-inner functions. Advances in Mathematics (2018).
2. Realization of functions on the symmetrized bidisc. Journal of Mathematical Analysis and Applications (2017).
3. Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc. Proceedings of the American Mathematical Society (2013).
4. The complex geometry of a domain related to μ-synthesis. Journal of Mathematical Analysis and Applications (2015).