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

By Ball J.A., Bolotnikov V.

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Extra info for A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space

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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. in: Topics in Interpolation Theory (Ed. H. Dym, B. Fritzsche, V. Katsnelson, and B. Kirstein), Operator Theory: Advances and Applications OT 95, pages 283–297. Birkh¨ auser Verlag, Basel, 1997. Translated from: Operators in function spaces and problems in function theory, pp. 83–96 (Naukova–Dumka, Kiev, 1987.

12) for some Schur function S ∈ Sd (E, E∗ ⊕ (⊕d1 E)): S(z) = E∗ S0 (z) : E→ , S1 (z) ⊕d1 E Z(z) = z1 IE ... zd IE . 13) Proof: The equivalence (1 ⇔ 2) follows from a more general fact that F is a contractive multiplier between two reproducing kernel Hilbert spaces H(K1 ) and H(K2 ) of functions analytic on a set Ω if and only if the kernel K2 (z, w) − F (z)K1 (z, w)F (w)∗ is positive on Ω. To show that (2 ⇔ 3), we represent KF as KF (z, w) = IE − F (z)F (w)∗ + F (z)Z(z)Z(w)∗ F (w)∗ , 1 − z, w or, equivalently, as KF (z, w) = A(z)A(w)∗ − B(z)B(w)∗ , 1 − z, w where A(z) = IE F (z)Z(z) and B(z) = F (z).

36] W. Rudin, F unction theory in the unit ball of Cn , Springer-Verlag, New York, 1980. [37] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, University of Arkansas Lecture Notes in the Matehmatical Sciences, Wiley, 1994. [38] D. Sarason, Nevanlinna-Pick interpolation with boundary data, Integral Equations and Operator Theory 30(2) (1998), 231-250. Joseph A.

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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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