Notes on the codimension one conjecture in the operator corona theorem

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 33-50

Résumé

Answering a question of S. R. Treil (2004), for every $\delta$, $0<\delta<1$, we constract examples of contractions whose characteristic function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ satisfies the conditions $\|F(z)x\|\geq\delta\|x\|$ and $\dim\mathcal E_\ast\ominus F(z)\mathcal E=1$ for every $z\in\mathbb D$, $x\in\mathcal E$, but is not left invertible. Also, we show that the condition $\sup_{z\in\mathbb D}\|I-F(z)^\ast F(z)\|_{\mathfrak S_1}<\infty$, where $\mathfrak S_1$ is the trace class of operators, which is sufficient for the left invertibility of the operator-valued function $F$ satisfying the estimate $\|F(z)x\|\geq\delta\|x\|$ for every $z\in\mathbb D$, $x\in\mathcal E$, with some $\delta>0$ (S. R. Treil, 2004), is necessary for the left invertibility of an inner function $F$ such that $\dim\mathcal E_\ast\ominus F(z)\mathcal E<\infty$ for some $z\in\mathbb D$.

Bibliographie

[1] J. R. Akeroyd, “Another look at some index theorems for the shift”, Indiana Univ. Math. J., 50 (2001), 705–718 | DOI | MR | Zbl

[2] J. R. Akeroyd, “A note on harmonic measure”, Comput. Methods Funct. Theory, 7:1 (2007), 91–104 | DOI | MR | Zbl

[3] A. Aleman, S. Richter, C. Sundberg, “Invariant subspaces for the backward shift on Hilbert spaces of analytic functions with regular norm”, Bergman spaces and related topics in complex analysis, Contemp. Math., 404, AMS, Providence, RI, 2006, 1–25 | DOI | MR | Zbl

[4] A. Aleman, S. Richter, C. Sundberg, “Nontangential limits in $\mathcal P^t(\mu)$-spaces and the index of invariant subspaces”, Ann. of Math., 169 (2009), 449–490 | DOI | MR | Zbl

[5] E. Kollingvud, A. Lovater, Teoriya predelnykh mnozhestv, Mir, M., 1971

[6] J. B. Conway, The theory of subnormal operators, Math. Surveys and Monographs, 36, Amer. Math. Soc., 1991 | DOI | MR | Zbl

[7] M. F. Gamal', “On contractions that are quasiaffine transforms of unilateral shifts”, Acta Sci. Math. (Szeged), 74 (2008), 755–765 | MR

[8] M. F. Gamal, “O czhatiyakh s kompaktnymi defektami”, Zap. nauchn. semin. POMI, 366, 2009, 13–41

[9] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, 199, New York, 2000 | DOI | MR | Zbl

[10] V. V. Kapustin, A. V. Lipin, “Operatornye algebry i reshetki invariantnykh podprostranstv. I”, Zap. nauchn. semin. LOMI, 178, 1989, 23–56 ; “II”, Зап. научн. семин. ЛОМИ, 190, 1991, 110–147 | MR | Zbl | MR | Zbl

[11] L. Kérchy, “Isometric asymptotes of power bounded operators”, Indiana Univ. Math. J., 38 (1989), 173–188 | DOI | MR | Zbl

[12] T. L. Miller, R. C. Smith, “Nontangential limits of functions in some $P^2(\mu)$ spaces”, Indiana Univ. Math. J., 39 (1990), 19–26 | DOI | MR | Zbl

[13] T. L. Miller, W. Smith, L. Yang, “Bounded point evaluations for certain $P^t(\mu)$ spaces”, Illinois J. Math., 43 (1999), 131–150 | MR | Zbl

[14] N. K. Nikolski, Operators, functions, and systems: an easy reading, v. I, Math. Surveys and Monographs, 92, Hardy, Hankel, and Toeplitz, AMS, 2002 | Zbl

[15] B. Sz.-Nagy, C. Foias, “On the structure of intertwining operators”, Acta Sci. Math. (Szeged), 35 (1973), 225–254 | MR | Zbl

[16] B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic analysis of operators on Hilbert space, Springer, New York, 2010 | MR | Zbl

[17] K. Takahashi, “On quasiaffine transforms of unilateral shifts”, Proc. Amer. Math. Soc., 100 (1987), 683–687 | MR | Zbl

[18] J. E. Thomson, L. Yang, “Invariant subspaces with the codimension one property in $L^t(\mu)$”, Indiana Univ. Math. J., 44 (1995), 1163–1173 | DOI | MR | Zbl

[19] V. A. Tolokonnikov, “Otsenki v teoreme Karlesona o korone, idealy algebry $H^\infty$, zadacha Sekefalvi–Nadya”, Zap. nauchn. semin. LOMI, 113, 1981, 178–198 | MR | Zbl

[20] S. R. Treil, “Ugly mezhdu koinvariantnymi podprostranstvami i operatornaya problema korony. Zadacha Sëkefalvi–Nadya”, Dokl. AN SSSR, 302:5 (1988), 1063–1068 | Zbl

[21] S. Treil, “Geometric methods in spectral theory of vector-valued functions: Some recent results”, Toeplitz operators and spectral function theory, Oper. Theory, Adv. Appl., 42, 1989, 209–280 | DOI | MR | Zbl

[22] S. Treil, “An operator corona theorem”, Indiana Univ. Math. J., 53 (2004), 1763–1781 | DOI | MR | Zbl

[23] S. Treil, “Lower bounds in the matrix corona theorem and the codimension one conjecture”, Geom. Funct. Anal., 14 (2004), 1118–1133 | DOI | MR | Zbl

[24] M. Uchiyama, “Contractions and unilateral shifts”, Acta Sci. Math. (Szeged), 46 (1983), 345–356 | MR | Zbl

[25] M. Uchiyama, “Curvatures and similarity of operators with holomorphic eigenvectors”, Trans. Amer. Math. Soc., 319 (1990), 405–415 | DOI | MR | Zbl