zbMATH — the first resource for mathematics

Similarity of operators and geometry of eigenvector bundles. (English) Zbl 1190.47002
From the authors’ introduction: “The main objects of this paper are operators with complete analytic families of eigenvectors, the backward shift being one of the simplest examples of such operators. Characterization of such operators up to unitary equivalence was completely done by M. J. Cowen and R. G. Douglas in [Acta Math. 141, 187–261 (1978; Zbl 0427.47016)].”
Let \(H\) be a Hilbert space and let \(T\in \mathcal B(H)\) be a contraction. Assume further that \(\dim\text{Ker}\,(T-\lambda I)=n<\infty\) for all \(\lambda\) in the unit disk \(\mathbb D,\) that \(\text{span}\,\cup_{\lambda \in \mathbb D}\text{Ker}\,(T-\lambda I)=H\) and that \(\text{Ker}\,(T-\lambda I)\) depends analytically on \(\lambda\).
On the vector Hardy space \(H_n^2\), the backward shift \(S^*_nB\) is defined in a similar way as is defined on \(H^2\); namely, \(S^*_nf(z)=\frac{f(z)-f(0)}{z}\).
For a contraction \(T\) as above, the authors prove that four statements are equivalent. The precise form of (2), (3) and (4) is very technical, but roughly, one finds:
(1) \(T\) is similar to \(S^*_n\).
(2) The eigenvector bundles of \(T\) and \(S_n^*\) are “uniformly equivalent”.
(3) The Green potential of an expression containing the mean curvature of the eigenvector bundle of \(T\) is uniformly bounded inside the disk \(\mathbb D\).
(3) A measure related to the mean curvature of the eigenvector bundle of \(T\) is a Carleson measure.
The equivalence of (1) and (2) was obtained by M. Uchiyama [Trans. Am. Math. Soc. 319, No. 1, 405–415 (1990; Zbl 0733.47015)] who also considered quasisimilarity, while B. Sz.-Nagy and C. Foiaş had characterized operators similar to isometries in terms of their characteristic functions in [Acta Sci. Math. 26, 79–91 (1965; Zbl 0138.38905)].

47A10 Spectrum, resolvent
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
30H10 Hardy spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: DOI Euclid EuDML arXiv
[1] M. Andersson, The corona theorem for matrices, Math. Z. 201(1) (1989), 121\Ndash130. · Zbl 0648.30039
[2] J.-P. Aubin, “Applied functional analysis” , Translated from the French by Carole Labrousse, With exercises by Bernard Cornet and Jean-Michel Lasry, John Wiley & Sons, New York-Chichester-Brisbane, 1979.
[3] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547\Ndash559. · Zbl 0112.29702
[4] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141(3-4) (1978), 187\Ndash261. · Zbl 0427.47016
[5] P. Griffiths and J. Harris, “Principles of algebraic geometry” , Reprint of the 1978 original, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. · Zbl 0836.14001
[6] N. K. Nikolski, “Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz” , Translated from the French by Andreas Hartmann, Mathematical Surveys and Monographs 92 , American Mathematical Society, Providence, RI, 2002. · Zbl 1007.47001
[7] N. K. Nikolski, “Operators, functions, and systems: an easy reading. Vol. 2. Model operators and systems” , Translated from the French by Andreas Hartmann and revised by the author, Mathematical Surveys and Monographs 93 , American Mathematical Society, Providence, RI, 2002. · Zbl 1007.47002
[8] N. K. Nikolski, “Treatise on the shift operator. Spectral function theory” , With an appendix by S. V. Hruščev and V. V. Peller, Translated from the Russian by Jaak Peetre, Grundlehren der Mathematischen Wissenschaften 273 , Springer-Verlag, Berlin, 1986.
[9] B. Sz.-Nagy and C. Foiaş, “Harmonic analysis of operators on Hilbert space” , Translated from the French and revised North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. · Zbl 0201.45003
[10] B. Sz.-Nagy and C. Foiaş, On the structure of intertwining operators, Acta Sci. Math. (Szeged) 35 (1973), 225\Ndash254. · Zbl 0272.47010
[11] B. Sz.-Nagy and C. Foiaş, Sur les contractions de l’espace de Hilbert. X. Contractions similaires à des transformations unitaires, Acta Sci. Math. (Szeged) 26 (1965), 79\Ndash91. · Zbl 0138.38905
[12] S. R. Treil, Angles between co-invariant subspaces, and the operator corona problem. The Szökefalvi-Nagy problem, (Russian), Dokl. Akad. Nauk SSSR 302(5) (1988), 1063\Ndash1068; translation in: Soviet Math. Dokl. 38(2) (1989), 394\Ndash399. · Zbl 0687.47004
[13] S. R. Treil, Geometric methods in spectral theory of vector-valued functions: some recent results, in: “Toeplitz operators and spectral function theory” , Oper. Theory Adv. Appl. 42 , Birkhäuser, Basel, 1989, pp. 209\Ndash280. · Zbl 0699.47009
[14] S. R. Treil, Unconditional bases of invariant subspaces of a contraction with finite defects, Indiana Univ. Math. J. 46(4) (1997), 1021\Ndash1054. · Zbl 0906.47005
[15] S. R. Treil, An operator Corona theorem, Indiana Univ. Math. J. 53(6) (2004), 1763\Ndash1780. · Zbl 1079.30049
[16] S. R. Treil, Lower bounds in the matrix Corona theorem and the codimension one conjecture, Geom. Funct. Anal. 14(5) (2004), 1118\Ndash1133. · Zbl 1057.47014
[17] S. R. Treil and B. D. Wick, Analytic projections, corona problem and geometry of holomorphic vector bundles, J. Amer. Math. Soc. 22(1) (2009), 55\Ndash76. · Zbl 1206.30076
[18] M. Uchiyama, Curvatures and similarity of operators with holomorphic eigenvectors, Trans. Amer. Math. Soc. 319(1) (1990), 405\Ndash415. · Zbl 0733.47015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.