Certain non-homogeneous matricial domains and Pick-Nevanlinna interpolation problem. (English) Zbl 1478.32046
Summary: In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows – generalizing a result of T. J. Ransford and M. C. White [Bull. Lond. Math. Soc. 23, No. 3, 256–262 (1991; Zbl 0749.32018)] for the spectral unit ball – that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider \(2\)-point and \(3\)-point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by G. Bharali [Integral Equations Oper. Theory 59, No. 3, 329–343 (2007; Zbl 1144.47013)] and V. S. Chandel [J. Geom. Anal. 30, No. 1, 551–572 (2020; Zbl 1436.30032)] related to these problems.
MSC:
| 32H35 | Proper holomorphic mappings, finiteness theorems |
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 30E05 | Moment problems and interpolation problems in the complex plane |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 32F45 | Invariant metrics and pseudodistances in several complex variables |
| 47A60 | Functional calculus for linear operators |
Keywords:
symmetrized product; invariant pseudo-distances; holomorphic automorphisms and proper maps; spectral unit ball; holomorphic functional calculus; Pick-Nevanlinna interpolation problemReferences:
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