Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Amar, Eric; Hartmann, Andreas

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 1871-1903.

Nous étudions des relations entre l’uniforme minimalité, l’inconditionnalité et l’interpolation pour des familles de noyaux reproduisants dans des espaces invariants par l’adjoint du shift. Cette classe d’espaces contient en particulier les espaces de Paley-Wiener pour lesquels il est connu que l’uniforme minimalité n’entraîne en général pas l’inconditionalité. Par conséquent, et contrairement à la situation dans les espaces de Hardy habituels (et dans d’autres échelles d’espaces), il semble nécessaire de changer la taille de l’espace afin de déduire l’inconditionnalité (ou l’interpolation) de l’uniforme minimalité. Un tel changement de la taille de l’espace peut être opéré de deux façons différentes : en diminuant l’exposant d’intégration, ou en “augmentant” la fonction définissante de l’espace (ce qui revient à augmenter le type dans le cas des espaces de Paley-Wiener). Les inégalités de Khinchin jouent un rôle central dans les preuves de nos résultats principaux.

We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and of other scales of spaces), changing the size of the space seems necessary to deduce unconditionality or interpolation from uniform minimality. Such a change can take two directions: lowering the power of integration, or “increasing” the defining inner function (e.g. increasing the type in the case of Paley-Wiener space). Khinchin’s inequalities play a substantial role in the proofs of our main results.

Classification : 30D55, 30E05, 46B09
Keywords: Uniform minimality, unconditional bases, model spaces, Paley-Wiener spaces, interpolation, one-component inner functions
Mot clés : uniforme minimalité, bases inconditionnelles, espaces modèles, espaces de Paley-Wiener, interpolation, fonctions intérieures à une composante

[1] Aleksandrov, A. B. On embedding theorems for coinvariant subspaces of the shift operator. II, J. Math. Sci., New York, Volume 110 (1999) no. 5, pp. 2907-2929 | DOI | MR | Zbl

[2] Amar, Eric On interpolation of interpolating sequences, Indag. Math. (N.S.), Volume 18 (2007) no. 2, pp. 177-187 | DOI | MR | Zbl

[3] Amar, Eric On linear extension for interpolating sequences, Stud. Math., Volume 186 (2008) no. 3, pp. 251-265 | DOI | MR

[4] Beurling, Arne The collected works of Arne Beurling. Volume 1: Complex analysis. Volume 2: Harmonic analysis. Ed. by Lennart Carleson, Paul Malliavin, John Neuberger, John Wermer, Contemporary Mathematicians. Boston etc.: Birkhäuser Verlag. xx, 475 p./v.1; xx, 389 p./v.2, 1989 | MR | Zbl

[5] Boricheva, Inna Geometric properties of projections of reproducing kernels on $z^{*}$-invariant subspaces of $H^2$, J. Funct. Anal., Volume 161 (1999) no. 2, pp. 397-417 | DOI | MR | Zbl

[6] Carleson, Lennart An interpolation problem for bounded analytic functions, Amer. J. Math., Volume 80 (1958), pp. 921-930 | DOI | MR | Zbl

[7] Carleson, Lennart Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2), Volume 76 (1962), pp. 547-559 | DOI | MR | Zbl

[8] Cohn, Bill Carleson measures for functions orthogonal to invariant subspaces, Pacific J. Math., Volume 103 (1982) no. 2, pp. 347-364 http://projecteuclid.org/getRecord?id=euclid.pjm/1102723968 | MR | Zbl

[9] Flornes, Kristin M. Sampling and interpolation in the Paley-Wiener spaces $L_{\pi }^p,0<p\le 1$, Publ. Mat., Volume 42 (1998) no. 1, pp. 103-118 | MR | Zbl

[10] Hartmann, Andreas; Sarason, Donald; Seip, Kristian Surjective Toeplitz operators, Acta Sci. Math. (Szeged), Volume 70 (2004) no. 3-4, pp. 609-621 | MR | Zbl

[11] Hruščëv, S. V.; Nikolskiĭ, N. K.; Pavlov, B. S. Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980) (Lecture Notes in Math.), Volume 864, Springer, Berlin, 1981, pp. 214-335 | MR | Zbl

[12] Levin, B. Ya. Lectures on entire functions, Translations of Mathematical Monographs, 150, American Mathematical Society, Providence, RI, 1996 (In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko) | MR | Zbl

[13] Lindenstrauss, J.; Zippin, M. Banach spaces with a unique unconditional basis, J. Functional Analysis, Volume 3 (1969), pp. 115-125 | DOI | MR | Zbl

[14] Lindenstrauss, Joram; Tzafriri, Lior Classical Banach spaces. I: Sequence spaces. II. Function spaces. Repr. of the 1977 and 1979 ed., Classics in Mathematics. Berlin: Springer-Verlag. xx, 432 p., 1996 | MR | Zbl

[15] Lyubarskii, Yurii I.; Seip, Kristian Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt’s $\left(A_p\right)$ condition, Rev. Mat. Iberoamericana, Volume 13 (1997) no. 2, pp. 361-376 | MR | Zbl

[16] Minkin, A. M. The reflection of indices and unconditional bases of exponentials, Algebra i Analiz, Volume 3 (1991) no. 5, pp. 109-134 | MR | Zbl

[17] Nikolskiĭ, N. K. Bases of invariant subspaces and operator interpolation, Trudy Mat. Inst. Steklov., Volume 130 (1978), p. 50-123, 223 (Spectral theory of functions and operators) | MR | Zbl

[18] Nikolskiĭ, N. K. Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 273, Springer-Verlag, Berlin, 1986 (Spectral function theory, With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre) | MR | Zbl

[19] Nikolskiĭ, N. K. Operators, functions, and systems: an easy reading, Mathematical Surveys and Monographs, 92 et 93, American Mathematical Society, Providence, RI, 2002 (Vol. 1, Hardy, Hankel and Toeplitz. Vol. 2, Model operators and systems, Translated from the French by Andreas Hartmann and revised by the author) | Zbl

[20] Pełczyński, A. Projections in certain Banach spaces, Studia Math., Volume 19 (1960), pp. 209-228 | MR | Zbl

[21] Rochberg, Richard Toeplitz operators on weighted $H^p$ spaces, Indiana Univ. Math. J., Volume 26 (1977) no. 2, pp. 291-298 | DOI | MR | Zbl

[22] Rosenthal, Haskell P. On the subspaces of $L^p$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math., Volume 8 (1970), pp. 273-303 | DOI | MR | Zbl

[23] Schuster, Alexander P.; Seip, Kristian A Carleson-type condition for interpolation in Bergman spaces, J. Reine Angew. Math., Volume 497 (1998), pp. 223-233 | DOI | MR | Zbl

[24] Schuster, Alexander P.; Seip, Kristian Weak conditions for interpolation in holomorphic spaces, Publ. Mat., Volume 44 (2000) no. 1, pp. 277-293 | MR | Zbl

[25] Seip, Kristian On the connection between exponential bases and certain related sequences in $L^2(-\pi ,\pi )$, J. Funct. Anal., Volume 130 (1995) no. 1, pp. 131-160 | DOI | MR | Zbl

[26] Shapiro, H. S.; Shields, A. L. On some interpolation problems for analytic functions, Amer. J. Math., Volume 83 (1961), pp. 513-532 | DOI | MR | Zbl

[27] Singer, Ivan Bases in Banach spaces. I, Springer-Verlag, New York, 1970 (Die Grundlehren der mathematischen Wissenschaften, Band 154) | MR | Zbl

[28] Treil, S. R.; Volberg, A. L. Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator, Algebra i Analiz, Volume 7 (1995) no. 6, pp. 205-226 | MR | Zbl