Abstract
This paper is concerned with the study of the singular values of a “four block operator” which naturally appears in control engineering and which possesses a number of interesting mathematical properties. The study of this operator will be shown to be reducible to a skew Toeplitz operator problem of the kind studied in [1]. The main theoretical fact proven here is an explicit closed form formula for the essential norm of the four block operator (see Proposition 1.1). We dedicate this paper to the memory of Constantin Apostol who was a great master of the essential properties of operators in Hilbert space.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H. Bercovici, C. Foias, and A. Tannenbaum, On skew Toeplitz operators, I, to appear in Integral Equations and Operator Theory.
J. Doyle, Lecture Notes, ONR/Honeywell Workshop on Advances in Multivariable Control, Minneapolis, Minnesota, 1984.
H. Dym and I. Gohberg, A new class of contractive interpolants and maximal entropy principles, to appear in Integral Equations and Operator Theory.
C. Foias and A. Tannenbaum, On the Nehari problem for a certain class of L ∞ functions appearing in control theory, J. Functional Analysis 74 (1987), pp. 146–159.
C. Foias, A. Tannenbaum, and G. Zames, On the H ∞-optimal sensitivity problem for systems with delays, SIAM J. Control and Optimization 25 (1987), 686–706.
C. Foias, A. Tannenbaum, and G. Zames, Some explicit formulae for the singular values of certain Hankel operators with factorizable symbol, Technical Report, Department of Electrical Engineering, Univ. of Minnesota, March 1987. Submitted for publication.
B.A. Francis, “A Course in H ∞ Control Theory,” Lecture Notes in Control and Information Science, Springer, New York, 1987.
E. Jonckheere and M. Verma, A spectral characterization of H ∞ optimal performance and its efficient computation, Systems and Control Letters 8 (1986), pp. 13–22.
N. K. Nikolskii, “Treatise on the Shift Operator,” Springer, New York, 1986.
B. Sz.-Nagy and C. Foias, “Harmonic Analysis of Operators on Hilbert Space,” North-Holland, Amsterdam, 1970.
G. Zames and S. Mitter, On Hankel + Toeplitz operators, MTNS Conference, Phoenix, Arizona, 1987.
G. Zames, A. Tannenbaum, and C. Foias, Optimal H ∞ interpolation: a new approach, Proceedings of the CDC, Athens, Greece, 1986, pp. 350-355.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Constantin Apostol
Rights and permissions
Copyright information
© 1988 Springer Basel AG
About this chapter
Cite this chapter
Foias, C., Tannenbaum, A. (1988). On the Four Block Problem, I. In: Gohberg, I. (eds) Topics in Operator Theory. Operator Theory: Advances and Applications, vol 32. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5475-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5475-7_8
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5477-1
Online ISBN: 978-3-0348-5475-7
eBook Packages: Springer Book Archive