The Lempert function of the symmetrized polydisc in higher dimensions is not a distance

Nikolov, Nikolai; Pflug, Peter; Zwonek, Włodzimierz

doi:10.1090/s0002-9939-07-08817-x

Cited by 33 publications

(22 citation statements)

References 13 publications

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“…The rich and surprising geometry of this domain and its higher-dimensional analogues has subsequently been elaborated by many authors (for example, [12,18,20,25,26]).…”

Section: Introductionmentioning

confidence: 99%

Extremal holomorphic maps and the symmetrized bidisc

Agler

Lykova

Young

2012

Proceedings of the London Mathematical Society

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We introduce the class of n‐extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. We show that a well‐known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence 𝒞ν, where ν⩾0, of necessary conditions for solvability, prove that they are of strictly increasing strength and show that 𝒞n−3 is insufficient for the solvability of an n‐point problem for n⩾3. We propose the conjecture that condition 𝒞n−2 is necessary and sufficient for the solvability of an n‐point interpolation problem for Γ and we explore the implications of this conjecture. We introduce a classification of rational Γ‐inner functions, that is, analytic functions from the disc into Γ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of Γ. The classes are related to n‐extremality and the conditions 𝒞ν; we prove numerous strict inclusions between the classes.

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“…The rich and surprising geometry of this domain and its higher-dimensional analogues has subsequently been elaborated by many authors (for example, [12,18,20,25,26]).…”

Section: Introductionmentioning

confidence: 99%

Extremal holomorphic maps and the symmetrized bidisc

Agler

Lykova

Young

2012

Proceedings of the London Mathematical Society

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“…Remark 5.2. The estimate above is better (especially asymptotically) than the general one from [9] (which is (n − 1)/n).…”

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confidence: 80%

Green functions of the spectral ball and symmetrized polydisk

Thomas

Trao

Zwonek

2011

Journal of Mathematical Analysis and Applications

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The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole matrix is derogatory, the inequality is always strict, and the difference between the two functions depends on the multiplicity of the eigenvalues as roots of the minimal polynomial of that matrix. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some estimates are given for invariant functions in the symmetrized polydisc, e.g. (infinitesimal versions of) the Carathéodory distance and the Green function, that show that they are distinct in dimension greater or equal to 3.

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“…It remains to use Proposition 2.2 with G = E E = G r 1 , D = E E and the fact that l G r 1 is not a distance (cf. [25]). 2…”

Section: Geometry Of the Generalized Tetrablockmentioning

confidence: 99%

Geometric properties of domains related to μ-synthesis

Zapałowski

2015

Journal of Mathematical Analysis and Applications

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In the paper we study the geometric properties of a large family of domains, called the generalized tetrablocks, related to the μ-synthesis, containing both the family of the symmetrized polydiscs and the family of the μ 1,n -quotients E n , n ≥ 2, introduced recently by G. Bharali. It is proved that the generalized tetrablock cannot be exhausted by domains biholomorphic to convex ones. Moreover, it is shown that the Carathéodory distance and the Lempert function are not equal on a large subfamily of the generalized tetrablocks, containing i.a. E n , n ≥ 4. We also derive a number of geometric properties of the generalized tetrablocks as well as the μ 1,n -quotients. As a by-product, we get that the pentablock, another domain related to the μ-synthesis problem introduced recently by J. Agler, Z.A. Lykova, and N.J. Young, cannot be exhausted by domains biholomorphic to convex ones.

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The Lempert function of the symmetrized polydisc in higher dimensions is not a distance

Abstract: Abstract. We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.

Cited by 33 publications

References 13 publications

Extremal holomorphic maps and the symmetrized bidisc

Extremal holomorphic maps and the symmetrized bidisc

Green functions of the spectral ball and symmetrized polydisk

Geometric properties of domains related to μ-synthesis

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