Notes on interpolation in the generalized Schur class. II. Nudel’man’s problem. (English) Zbl 1060.47020
Trans. Am. Math. Soc. 355, No. 2, 813-836 (2003); corrigendum ibid. 371, No. 5, 3743-3745 (2019).
[For part I, see Oper. Theory, Adv. Appl. 134, 67–97 (2002; Zbl 1030.30036).]
Let \(\mathbf{S}_\kappa\) be the set of functions \(S(z)\) of the form \(S(z)=f(z)/B_\kappa(z)\), where \(B_\kappa(z)\) is a Blaschke product of degree \(\kappa\). \(\mathbf{S}_\kappa\) is called the generalized Schur class of degree \(\kappa\). The authors consider the following version of Nudel’man’s problem for \(\mathbf{S}_\kappa\): {Given a complex vector space \(V\), two vectors \(b,c\in V\), and a linear operator \(A\) on \(V\) into itself, find a pair \((f,B_\kappa)\), where \(f\in \mathbf{S}_0\), such that \(f(A)c=B_{\kappa}(A)b\).}
A condition for when the solution of the above problem can be found is given by the main theorem of the present paper. This result is applied in describing a unified approach to a series of known interpolation problems for the generalized Schur and Nevanlinna class functions with interior and boundary data.
Editor’s remark: A completely revised version of this paper has been published separately in [Complex Anal. Oper. Theory 14, No. 1, Paper No. 25, 30 p. (2020; Zbl 1435.47024)].
Let \(\mathbf{S}_\kappa\) be the set of functions \(S(z)\) of the form \(S(z)=f(z)/B_\kappa(z)\), where \(B_\kappa(z)\) is a Blaschke product of degree \(\kappa\). \(\mathbf{S}_\kappa\) is called the generalized Schur class of degree \(\kappa\). The authors consider the following version of Nudel’man’s problem for \(\mathbf{S}_\kappa\): {Given a complex vector space \(V\), two vectors \(b,c\in V\), and a linear operator \(A\) on \(V\) into itself, find a pair \((f,B_\kappa)\), where \(f\in \mathbf{S}_0\), such that \(f(A)c=B_{\kappa}(A)b\).}
A condition for when the solution of the above problem can be found is given by the main theorem of the present paper. This result is applied in describing a unified approach to a series of known interpolation problems for the generalized Schur and Nevanlinna class functions with interior and boundary data.
Editor’s remark: A completely revised version of this paper has been published separately in [Complex Anal. Oper. Theory 14, No. 1, Paper No. 25, 30 p. (2020; Zbl 1435.47024)].
Reviewer: Jesus Illán González (Vigo)
MSC:
| 47A57 | Linear operator methods in interpolation, moment and extension problems |
| 30E05 | Moment problems and interpolation problems in the complex plane |
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
| 47B50 | Linear operators on spaces with an indefinite metric |
| 42A50 | Conjugate functions, conjugate series, singular integrals |
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