A band method approach to a positive expansion problem in a unitary dilation setting

In this paper a positive commuting expansion problem is presented and solved. The problem is set up in the framework of a minimal unitary dilation of a contraction acting on a Hilbert space and includes the Carathéodory and other classical interpolation problems. By combining the geometry of the minimal unitary dilation with state space techniques from system theory, a special solution is constructed. Next using the band method approach and spectral factorizations of this special solution a linear fractional parameterization of all solutions is obtained. Explicit state space formulas and applications to some classical interpolation problems are given.

[1]  D. Sarason.  Generalized interpolation in , 1967 .

[2]  I. Gohberg,et al.  Extensions with positive real part. A new version of the abstract band method with applications , 1993 .

[3]  C. Foias,et al.  The commutant lifting approach to interpolation problems , 1990 .

[4]  B. Øksendal,et al.  Spaces of Analytic Functions , 1976 .

[5]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[6]  W. Rugh.  Linear System Theory , 1992 .

[7]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[8]  Marinus A. Kaashoek,et al.  The band method and generalized Carathéodory-Toeplitz interpolation at operator points , 1999 .

[9]  Israel Gohberg,et al.  The band method for positive and strictly contractive extension problems: An alternative version and new applications , 1989 .

[10]  C. Foias,et al.  Harmonic Analysis of Operators on Hilbert Space , 1970 .

[11]  Arthur E. Frazho,et al.  Metric Constrained Interpolation, Commutant Lifting and Systems , 1998 .

[12]  Thomas Kailath,et al.  Linear Systems , 1980 .

[13]  F. L. Auspitz.  Systems theory and design , 1988 .

[14]  P. Fillmore.  Notes on operator theory , 1970 .

[15]  The Band Method and Grassmannian Approach for Completion and Extension Problems , 1996 .

[16]  Tiberiu Constantinescu.  Schur Parameters, Factorization and Dilation Problems , 2002 .

[17]  K. Hoffman.  Banach Spaces of Analytic Functions , 1962 .

[18]  P. Halmos.  A Hilbert Space Problem Book , 1967 .