Abstract
A lifting theorem for representations of the algebra of block upper triangular matrices is proved analogous to the commutatant lifting theorem of Sarason and Sz.-Nagy and Foias. Included is a description by linear fractional maps of all solutions of the lifting problem. This first part of the paper contains the statements of the main results and applications to matrix interpolation and completion problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A] Abrahamse, M.B.: The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26(1979), 195–203.
[AD 1] Abrahamse, M.B. and Dourglas, R.G.: A class of subnormal operators related to multiply connected domains, Advances in Math. 19(1976), 106–148
[AD 2] Abrahamse, M.B. and Douglas, R.G.: Operators on multiply connected domains, Proc. Roy. Irish Acad. Ser. A 74(1974), 135–141.
[AAK] Adamjan, V.M., Arov, D.Z. and Krein, M.G.: Infinte Hankel block matrices and related extension problems, Amer. Math. Soc. Transl. (2) 111(1978), 133–156.
[Ag] Agler, J.: Rational dilation on an annulus, preprint.
[Ar 1] Arveson, W.B.: Subalgebras on C*-algebras I, II, Acta Math. 123 (1969), 141–224; 128(1972), 271–308.
[Ar 2] Arveson, W.B.: Interpolation problems in nest algebras, J. Functional Analysis 20(1975), 208–233.
[Ar 3] Arveson, W.B.: An Invitation to C*-algebras, Springer-Verlag, New York-Heildelberg-Berlin (1976).
[B 1] Ball, J.A.: Operators of class C∞ over multiply connected domains, Michigan Math. J. 25(1978), 183–195.
[B 2] Ball, J.A.: A lifting theorem for operator models of finite rank on multiply connected domains, J. Operator Theory 1(1979), 3–25.
[B 3] Ball, J.A.: Interpolation problems and Toeplitz operators on multiply connected domains, Integral Equations and Operator Theory 4(1981), 172–184.
[BG] Ball, J.A. and Gohberg, I.: Shift invariant subspaces, factorization and interpolation for matrices I: The canonical case, Linear Algebra and Applications, to appear.
[BH] Ball, J.A. and Helton, J.W.: A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory, J. Operator Theory 9(1983), 107–142.
[C] Carleson, L.: Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76(1962), 547–559.
[Cl 1] Clark, D.N.: Commutants that do not dilate, Proc. Amer. Math. Soc. 35(1972), 483–486.
[Cl 2] Clark, D.N.: On commuting contractions, J. Math. Anal. Appl. 32(1970), 590–596.
[Co] Conway, J.B.: Subnormal Operators, Pitman, Boston-London, Melbourne, (1981).
[DKW] Davis, C., Kahane, W.M. and Weinberger, H.F.: Norm preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal. 19(1982), 445–469.
[DMP] Douglas, R.G., Muhly, P.S. and Pearch, C.: Lifting commuting operators, Michigan Math. J. 15(1968), 385–395.
[DPl] Douglas, R.G. and Paulsen, V.I.: Completely bounded maps and hypo-Dirichlet algebras, preprint.
[DG] Dym, H. and Gohberg, I.: Extensions of band matrices with band inverses, Linear Algebra and its Applications 36(1981), 1–24.
[F] Feintuch, A.: Discrete-time feedback systems: an operator theoretic approach, Advances in Applied Math. 4(1983), 103–123.
[GJSW] Grone, R., Johnson, C.R., Sa, E.M. and Wolkowicz, H.: Positive definite completions of partial Hermitian matrices, Linear Algebra and Applications, 58(1984), 109–124.
[H] Halmos, P.R., Shifts on Hilbert space, J. Reine Angew. Math. 208 (1961), 102–112.
[McAMS] McAsey, M.: Muhly, P.S. and Saitoh, K.-S.: Nonselfadjoint crossed products (invariant subspaces and maximality), Trans. Amer. Math. Soc. 248(1979), 381–409.
[N] Sz.-Nagy, B.: Sur les contractions de l'espace de Hilbert, Acta Sci. Math. (Szeged) 15(1953), 87–92.
[NF] Sz.-Nagy, B. and Foias, C.: Harmonic Analysis of Operators on Hilbert Space, American Elsevier, New York (1970).
[Ne] Nehari, Z.: On bounded bilinear forms, Ann. of Math. (2) 65 (1957), 153–162.
[P] Page, L.: Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150(1970), 529–539.
[Pl] Paulsen, V.: private communication.
[Po] Power, S.C.: The distance to upper triangular operators, Math. Proc. Cambridge Philos. Soc. 88(1980), 327–329.
[R] Rosenblum, M.: A corona theorem for countably many functions, Integral Equations and Operator Theory 3(1980), 125–137.
[S] Sarason, D.: Generalized interpolation in H∞, Trnas. Amer. Math. Soc. 127(1967), 179–203.
Author information
Authors and Affiliations
Additional information
The work was partially supported by a grant from the U. S. National Science Foundation.
Rights and permissions
About this article
Cite this article
Ball, J.A., Gohberg, I. A commutant lifting theorem for triangular matrices with diverse applications. Integr equ oper theory 8, 205–267 (1985). https://doi.org/10.1007/BF01202814
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01202814