Helsingin Yliopisto


Toeplitz-operators on Bergman spaces

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Department of Mathematics and Statistics of University of Helsinki


Formula 1. Toeplitz-operator and Bergman projection

Bergman space Ap with 1 ≤ p < ∞ is the closed subspace of Lp consisting of analytic functions. We consider here the case the domain is the open unit disc D of the complex plane C and the Lp -space is defined with the normalized two-dimensional Lebesgue measure dA.
The Toeplitz-operator Ta is defined as the compose of the multiplication Ma and the Bergman projection P . Obviously, since P : Lp → Ap is well defined and bounded in case 1 < p < ∞ , any bounded symbol a defines a bounded Toeplitz-operator Ta : Ap → Ap . Nearly equally obviously, very badly singular functions (indeed, distributions) which are compactly supported in D, also define bounded Toeplitz-operators. A satisfactory characterization of the boundedness of the operator in terms of the symbol is a remarkable open problem.


  • J.Bonet, W.Lusky, J.Taskinen:
    On boundedness and compactness of Toeplitz operators in weighted H^\infty-spaces.
    To appear in J. Functional Anal.

    This is a study of Toeplitz operators with radial symbols on the unit disc in weighted H^\infty spaces. Assuming that the weight satisfies the condition (B) introduced by the second author, we are able to characterize the boundedness and compactness of these operators. Note that the corresponding characterization remains unknown in the Bergman spaces A^p with 1 < p < \infty, unless p = 2.
    One of the starting points is that for radial symbols, the Toeplitz operators are just Taylor coefficient multipliers, and this creates a link to our recent studies on solid hulls, although the results cannot be directly drawn from them.
    In addition, we formulate a related sufficient condition for the boundedness in reflexive weighted Bergman spaces.
    Moreover, it is known by results of M.Dostanic that the Bergman projection may behave badly in weighted Bergman spaces with p not equal to 2, if the weight is rapidly decreasing at the boundary of the disc. We complement this result by constructing a bounded harmonic symbol such that the corresponding Toeplitz operator is not bounded in any weighted H_v^\infty space with the weight v satisfying some mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms.

  • A.Karapetyants, J.Taskinen:
    Toeplitz operators with radial symbols on weighted holomorphic Orlicz spaces.

    We consider a class of Toeplitz operators with radial symbols on weighted holomorphic Orlicz space. Our result is a generalization of that in the paper by W. Lusky and the second author in Studia Math. 204 (2011), 137-154 (see below): we characterize the boundedness and compactness in the case some iterated radial integral of the symbol is positive.

  • J.Taskinen, J.Virtanen:
    On compactness of Toeplitz operators in Bergman spaces.
    Functiones Approximatio 59.2 (2018), 305-318.

    We give a characterization of compact Toeplitz operators in Bergman spaces A^p, 1 < p < \infty, of the unit disc with symbols in L^1 under a mild additional condition. Our result is new even in the Hilbert space setting p=2, where it extends the well-known characterization of compact Toeplitz operators with bounded symbols by Stroethoff and Zheng (Transactions AMS -92).

  • J.Taskinen, J.Virtanen:
    On generalized Toeplitz and little Hankel operators on Bergman spaces.
    Archiv Math. 110, 2 (2018), 155--166.

    This work complements the Rev.Mat.Iberoamericana-2010-paper of the authors (below). We find a concrete integral formula for the class of generalized Toeplitz operators, studied in the above citation. The result is extended to little Hankel operators. We show that the approximating series and limits for the generalized Toeplitz operators converge in the strong operator topology. We give an example of an L2-symbol a such that the Toeplitz operator with symbol |a| fails to be bounded, although the operator with symbol a can be seen seen to be bounded by using our generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.

  • A.Perälä, J.Taskinen, J.Virtanen:
    Toeplitz operators on Dirichlet-Besov spaces.
    Houston J. Math. 43, 1(2017), 93_-108.

    We study Toeplitz operators Ta on the Besov spaces Bp in the case of the open unit disk and finite p > 1 . We prove that a symbol a satisfying a weak Lipschitz type condition induces a bounded operator Ta . Such symbols do not need to be bounded functions or have continuous extensions to the boundary of unit disc. We discuss the problem of the existence of nontrivial compact Toeplitz operators and also consider Fredholm properties and prove an index formula.

  • J.Bonet, J.Taskinen:
    A note about Volterra operators on weighted Banach spaces of entire functions.
    Math. Nachrichten 288 (2015), 1216-1225.

    We characterize boundedness, compactness and weak compactness of Volterra operators Vg acting between different weighted Banach spaces H_v^\infty of entire functions with sup-norms in terms of the symbol g; thus we complement recent work by Bassallote, Contreras, Hernáandez-Mancera, Martín and Paul for spaces of holomorphic functions on the disc and by Constantin and Peláez for reflexive weighted Fock spaces.

  • A.Perälä, J.Taskinen, J.Virtanen:
    New results and open problems on Toeplitz operators in Bergman spaces.
    New York J. Math. 17a (2011), 147-164.

    We review recent developements of the theory of Toeplitz operators in Bergman spaces. In addition we consider the special case of radial distributional symbols on the disc. We reformulate the sufficient conditions of the Rev.Math.Iberoam- and P.E.M.S.-papers (both below) for this case and show that the latter condition is weaker than the former.
    There is also a short remark on the possibility to extend the definition of Toeplitz operators even beyond distributional symbols, i.e. to the case of hyperfunctions.

  • A.Perälä, J.Taskinen, J.Virtanen:
    Toeplitz operators with distributional symbols on Fock spaces.
    Funct. et approx. 44,2 (2011), 203-213.

    We show how Toeplitz operators with distributional symbols can be defined and treated in Fock spaces of entire analytic functions. The approach is a nontrivial adaptation of the P.E.M.S.-paper to the Fock space setting.

  • W.Lusky, J.Taskinen:
    Toeplitz operators on Bergman spaces and Hardy multipliers.
    Studia Math. 204 (2011), 137-154.

    The setting is Toeplitz operators Ta in weighted Bergman spaces on the unit disc; both the symbols and weights are assumed radial. In the Hilbert space case Grudsky, Karapetyants and Vasilevski showed that the Toeplitz operator is unitarily equivalent to the Taylor coefficient multiplier and used this approach to derive a variety of results for Ta.
    The same is not possible in the case of non-Hilbert Bergman spaces, since the monomials do not form an unconditional Schauder bases there. However, using the methods of the first named author, it is still possible to decompose the Bergman space into a sequence of finite dimensional blocks. The restriction of Ta onto these blocks can be still be considered as a coefficient multiplier, and this way we can derive a connection of the boundedness problem of Ta in the Bergman space to the boundedness problem of Hardy multipliers.
    In the other part of the paper we characterize the bounded Toeplitz operators under an additional weak assumption on the positivity of certain indefinite integrals of the symbol. This is a generalization of the well-known result on boundedness for positive symbols.

  • J.Taskinen, J.Virtanen:
    Weighted BMO and Toeplitz operators on the Bergman space .
    J.Operator Th. 68,1 (2012), 131-140.

    We adopt the results of the Rev.Math.Iberoam.-paper to the case p=1. In addition to an inevitable logarithmic correction, it is necessary to consider separately BA and BO type symbols.

  • J.Taskinen, J.Virtanen:
    Toeplitz operators on Bergman spaces with locally integrable symbols.
    Rev.Math.Iberoamericana. 26,2 (2010), 693-706.

    For positive symbols the boundedness of Ta: Ap → Ap, 1 < p < ∞, can be characterized by the boundedness of averages of the symbol a over all hyperbolic discs with some constant radius. It is quite obvious that the straightforward generalization of this condition for nonpositive a (by replacing a with |a|) is a sufficient but not necessary condition for boundedness of Ta: Ap → Ap.
    In the present work we give a much weaker sufficient condition, which is still a kind of averaging condition over hyperbolic rectangles (however, for a itself, rather than for |a|). We also give an example of a symbol which satisfies our condition but is not integrable over the unit disc.
    We also consider sufficient compactness and Fredholmness conditions, and prove a rather general index formula, which again uses an average function instead of a usual pointwise condition for a.

  • A.Perälä,J.Taskinen, J.Virtanen:
    Toeplitz operators with distributional symbols on Bergman spaces.
    Proc.Edinburgh Math.Soc. 54, 2 (2011), 505-514.

    It is not at all difficult to define Toeplitz-operators with distributional symbols for distributions which are compactly supported in D: the integral formula defining a usual Toeplitz operator can be seen as a duality bracket of a distribution and an infinitely smooth test function (which is the function f ∈ Ap times the Bergman kernel, cf. Formula 1).
    We give the definition of a general class of Toeplitz-operators with noncompactly supported distributions as symbols. The operators are well defined and bounded on the reflexive Bergman spaces, as long as the symbol belongs to a negative order weighted Sobolev space, the weight being the boundary distance to the power and α being the order of the distribution.
    The corresponding condition for compactness is also provided.

  • J.Taskinen, J.Virtanen:
    Spectral theory of Toeplitz and Hankel operators on the Bergman space A¹.
    New York J. Math. 14 (2008),1-19.

    In case p=1 the Bergman projection is not bounded, contrary to the reflexive cases. Consequently, the conditions sufficent for the boundedness of Toeplitz-operators on A₁ include some logarithmic factors in a form or another. Here, we show that bounded symbols which belong to the hyperbolic logarithmic BMO-space induce bounded Toeplitz-operators A₁→A₁. We also prove corresponding compactness and Fredholmness results and provide an index formula in terms of the winding number of the symbol.

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