Abstract
Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.
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References
[Ar] D. Arov, RegularJ-inner matrix-functions and related continuation problems,Operator Theory: Advances and Applications,OT43, Birkhäuser Verlag, Basel, 1990, pp. 63–87.
[Ba] J.A. Ball, Commutant lifting and interpolation: the time-varying case,Integral Equations and Operator Theory, 25 (1996), pp. 377–405.
[BG] J.A. Ball and I. Gohberg, A commutant lifting theorem for triangular matrices with diverse applications,Integral Equations and Operator Theory 8 (1985), 205–267.
[BGK] J.A. Ball, I. Gohberg and M.A. Kaashoek, Two-sided Nudelman interpolation for input-output operators of discrete time-varying systems,Integral Equations and Operator Theory 20 (1994), 174–211.
[Da] K. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series 191, Longman Scientific & Technical, Essex and Wiley, New York 1988.
[DD] P. Dewilde and H. Dym, Interpolation for upper triangular operators, in:Timevariant systems and interpolation (I. Gohberg, ed.),Operator Theory: Advances and Applications,OT56, Birkhäuser Verlag, Basel, 1992, pp. 153–260.
[DF] H. Dym and B. Freydin, Bitangential interpolation for upper triangular operators, to appear inOperator Theory: Advances and Applications.
[D] H. Dym, Review of “The Commutant Lifting Approach to Interpolation Problems” by C. Foias and A.E. Frazho,Bull. AMS. 31(1) (1994), 125–140.
[FF] C. Foias and A. Frazho, The Commutant Lifting Approach to Interpolation Problems, Birkhäuser Verlag, Basel, 1990.
[F] B. Freydin, Topics in time varying interpolation, Ph. D. Thesis, Rehovot (1996).
[K] A.Ya. Kheifets, The generalized bitangential Schur-Nevanlinna-Pick problem and the related Parseval equality, in:Journal of Soviet Mathematic,58, No. 4 (1992), 358–364.
[KY] A.Ya. Kheifets and P.M. Yuditskii, An analysis and extension of V.P. Potapov's approach to interpolation problems with applications to the generalized bitangential Schur-Nevanlinna-Pick problem and J-inner-outer factorization.Operator Theory: Advances and Application OT72 (1994), Birkhäuser Verlag, Basel.
[Ko] J. Kos, Time-dependent problems in linear operator theory, Ph. D. Thesis, Amsterdam (1995).
[PP] V.I. Paulsen and S. Power, Lifting theorem for nest algebras.J. Oper. Theory 20 (1988), 311–328.
[RR] M. Rosenblum and J. Rovnyak,Hardy Classes and Operator Theory, Oxford Univ. Press, New York, 1985.
[S1] D. Sarason, Invariant subspaces and unstarred operator algebras,Pacific J. Math. 17 (1966), 511–517.
[S2] D. Sarason, Generalized interpolation inH ∞,Trans. Amer. Math. Soc. 127 (1967), 179–201.
[V] A. van der Veen, Time-varying theory and computational modeling, Ph. D. Thesis, Delft (1993).
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Freydin, B. Commutant lifting theorem and interpolation in discrete nest algebras. Integr equ oper theory 29, 211–230 (1997). https://doi.org/10.1007/BF01191431
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DOI: https://doi.org/10.1007/BF01191431