Ball J.A., Bolotnikov V.'s A Bitangential Interpolation Problem on the Closed Unit Ball PDF

By Ball J.A., Bolotnikov V.

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R. Pitts, Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321– 337. [23] H. Dym, J contractive matrix functions, reproducing kernel spaces and interpolation, CBMS Lecture Notes, vol. 71, Amer. Math. , Rhodes Island, 1989. [24] G. Julia, Extension nouvelle d’un lemme de Schwarz, Acta Math. 42 (1920), 349–355. [25] V. Katsnelson, A. Kheifets, and P. Yuditskii. An abstract interpolation problem and the extension theory of isometric operators.

The choice T = 0 leads to the function S(z) = Σ11 (z). This function is analytic at β1 and β2 and it is readily seen that Σ11 (β2 ) = Σ11 ((0, 1)) = −1. Σ11 (β1 ) = Σ11 ((1, 0)) = 1 and Furthermore, 3r+4 1 − 11−4r 1 − |Σ11 (rβ1 )|2 lim = lim r→1 r→1 1 − r2 1 − r2 and 4−7r 1 − 11−8r 1 − |Σ11 (rβ2 )|2 = lim lim 2 r→1 r→1 1−r 1 − r2 Another choice of   T = √ 2 11 7 0 √ 5 7 2 =1 2 = 5. 4) to S(z) = (1 − z1 )(1 − z2 ) 5 + 5z1 z2 − 2z12 − 3z1 − 5z2 3z1 − 7z2 + 4 + · . z1 z2 −4z1 −8z2 +11 z1 z2 −4z1 −8z2 +11 5z1 z2 − 9z1 − 5z2 + 2z12 + 7 A simple computation shows that S(rβ1 ) = 2r + 3 , 7 − 2r S(rβ2 ) = 3 − 5r .

1) and 1 − |S(rβ1 )|2 4 = ≤ 1, 2 r→1 1−r 5 lim 1 − |S(rβ2 )|2 = 5. r→1 1 − r2 lim References [1] M. Abate, Angular derivatives in strongly pseudoconvex domains, Proc. Sympos. , 52 (1991), 23–40. [2] J. Agler and J. E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. , 175 (2000), 111–124. [3] N. G. Kre˘ın, Some questions in the theory of moments, Article II, Translations of Mathematical Monographs, Amer. Math. , 1962. [4] D. Alpay, V. Bolotnikov and T. Kaptano˘ glu, The Schur algorithm and reproducing kernel Hilbert spaces in the ball, Linear Algebra Appl.

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A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space by Ball J.A., Bolotnikov V.


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