Operator Theory and Classical Analysis



Operator identities in interpolation and spectral theory,

joint papers by J. Rovnyak and L. A. Sakhnovich



Canonical model and linear systems


  • L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics, Wiley, 1966, pp. 295-392
  • L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart, and Winston, New York, 1966
  • D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl. 96, Birkhauser, 1997

Related works:
D. Z. Arov and H. Dym, J-Contractive matrix valued functions and related topics, Cambridge University Press, 2008.
J. A. Ball and V. Bolotnikov, de Branges-Rovnyak spaces: basics and theory, Operator Theory, Vol.~1, 631-679, Edited by Daniel Alpay, Springer, Basel (2015); de Branges-Rovnyak spaces and norm-constrained interpolation, ibid. 681-720.
L. de Branges, Krein spaces of analytic functions, J. Funct. Anal. 81 (1988), 219-259.
E. Fricain and J. Mashreghi, The theory of H(b) spaces, Vols. 1, 2, Cambridge University Press, 2016.


Odds and ends


  • J. Rovnyak, An extension problem for the coefficients of Riemann mappings, University of Virginia seminar lecture, November 1991. The topic of the lecture is a theorem of L. de Branges on the Taylor coefficients of a univalent function. The lecture is an exposition of an elegant result that I learned from a book draft that L. de Branges circulated in the 1980s (unpublished); the result is quoted without proof as Theorem 1.2 in J. Rovnyak, Coefficient estimates for Riemann mapping functions, J. Analyse Math. 52 (1989), 53-93. V. I. Vasyunin and N. K. Nikolskii prove the result in pp. 1219-1225 of their paper, Operator-valued measures and coefficients of univalent functions, St. Petersburg Math. J. 3 (1992), pp. 1199-1270; statements by these authors on pp. 1203, 1219 cast doubt on the completeness of the original proof of de Branges. I believe the original proof of de Branges is complete and correct, and my lecture fleshes out the details.
  • J. Rovnyak, Characterization of spaces H(M), Unpublished paper, 1968. I thank D. Z. Arov and H. Dym for citing this paper in their book, J-Contractive Matrix Valued Functions and Related Topics, Cambridge University Press, 2008, pp. 254, 332.