Abstract
Robust feedback control ideas crystalized in the 1980’s under the form of H ∞ control. An important issue in this theory is the effective calculation of the norms of certain operators. In many cases of interest these operators can be written as (scalar or more general) functions of a given contraction T,and this makes it possible to bring into the picture ideas from the dilation theory of contractions. These ideas were used for the first time in [7] for the calculation of ‖ f (T)‖, where f is a rational function, and T is a contraction with defect indices equal to one. A more general approach was introduced in Part I of this paper [2]. In [2] the operator T was allowed to have finite defect indices, and the calculation of ‖f (T)‖ (with f no longer a scalar function) was replaced by the study of invertibility for skew Toeplitz operators. The work in [2] was given an explicitly algorithmic form for scalar Toeplitz operators in [6]. A unified presentation of these results is given in [3].
The authors were partially supported by grant from the National Science Foundation, Air Force Office of Scientific Research, and Army Research Office.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Bercovici, Operator Theory and Arithmetic in H ∞, Math. Surveys and Monographs No, 56, Amer. Math. Soc., Providence, Rhode Island, 1988.
H. Bercovici, C. Foias, and A. Tannenbaum, On skew Toeplitz operators. I, Operator Theory: Advances and Applications 29(1988), 21–43.
C. Foias, Commutant lifting techniques for computing H ∞ controlers. In: ‘H∞-control Theory,’ Lecture Notes in Mathematics, No. 1496. Springer-Verlag, New York, 1991.
C. Foias and A. Frazho, The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, Vol. 44. Birkhäuser, Boston, 1990.
C. Foias and B. Sz.-Nagy, Harmonic Analysis of Operators in Hilbert Space. North Holland, Amsterdam, 1970.
C. Foias and A. Tannenbaum, Some remarks on optimal interpolation, System and Control Letters 11(1988), 259–265.
C. Foias, A. Tannenbaum, and G. Zames, On the H ∞ -optimal sensitivity problem for systems with delays, SIAM J. Control and Optimization 25(1987), 686–705.
C. Gu, Eliminating the genericity conditions in the skew Toeplitz operator algorithm, SIAM J. Math. Anal. 23(1992), 1623–1638.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Basel AG
About this chapter
Cite this chapter
Bercovici, H., Foias, C., Tannenbaum, A. (1998). On Skew Toeplitz Operators, II. In: Bercovici, H., Foias, C.I. (eds) Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics. Operator Theory Advances and Applications, vol 104. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8779-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8779-3_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9771-6
Online ISBN: 978-3-0348-8779-3
eBook Packages: Springer Book Archive