Estimates of the Carathéodory metric on the symmetrized polydisc

Nikolov, Nikolai; Pflug, Peter; Thomas, Pascal J.; Zwonek, Włodzimierz

doi:10.1016/j.jmaa.2007.09.072

Cited by 15 publications

(20 citation statements)

References 13 publications

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“…Even though we cannot prove the above conjecture, we are able to get some estimates between (logarithm of) the Carathéodory distance and the Green function in the symmetrized polydisc, showing in particular that these two objects differ in G n , n ≥ 3, which extends some of the results from [10]. We get this from facts about their infinitesimal versions.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 69%

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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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Section: Introduction and Statement Of Resultssupporting

confidence: 69%

“…This part of the paper may be seen as a continuation and extension of the results from [10]. Recall [5] that for any k ∈ Z * + , X).…”

mentioning

confidence: 86%

Green functions of the spectral ball and symmetrized polydisk

Thomas

Trao

Zwonek

2011

Journal of Mathematical Analysis and Applications

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“…Our argument follows from a recent study [11] of the Carathéodory metric on G n , n ≥ 3. This argument is presented in the next section.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Some New Observations on Interpolation in the Spectral Unit Ball

Bharali

2007

Integr. equ. oper. theory

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Abstract. We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ωn, n ≥ 2. We begin by showing that a known necessary condition for the existence of a O(D; Ωn)-interpolant (D here being the unit disc in C), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem -one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ωn, n ≥ 2.

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“…It remains to use Proposition 2.2 with G = E E = G 3 , D = E E and the fact that c G 3 (0, ·) ≡ k G 3 (0, ·) (cf. [24]). 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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Estimates of the Carathéodory metric on the symmetrized polydisc

Abstract: Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.

Cited by 15 publications

References 13 publications

Green functions of the spectral ball and symmetrized polydisk

Green functions of the spectral ball and symmetrized polydisk

Some New Observations on Interpolation in the Spectral Unit Ball

Geometric properties of domains related to μ-synthesis

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