Let H ∞ denote the space of bounded analytic functions on the upper half-plane ℂ+. We prove that each function in the model space H ∞ ∩ Θ\( \overline{H^{\infty }} \) is an operator Lipschitz function on ℝ if and only if the inner function Θ is a usual Lipschitz function, i.e., Θ′ ∈ H ∞. Let (OL)′(ℝ) denote the set of all functions f ∈ L ∞ whose antiderivative is operator Lipschitz on the real line ℝ. We prove that H ∞ ∩ Θ\( \overline{H^{\infty }} \) ⊂ (OL)′(ℝ) if Θ is a Blaschke product with zeros satisfying the uniform Frostman condition. We also deal with the following questions. When does an inner function Θ belong to (OL)′(ℝ)? When does each divisor of an inner function Θ belong to (OL)′(ℝ)? As an application, we deduce that (OL)′(ℝ) is not a subalgebra of L ∞(ℝ). Another application is related to a description of the sets of discontinuity points for the derivatives of operator Lipschitz functions. We prove that a set ℰ,ℰ ⊂ ℝ, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if ℰ is an F σ set of first category. A considerable proportion of the results of the paper is based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton, and Friedman. We also give a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition. Bibliography: 27 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. R. Ahern and D. N. Clark, “Radial Nth derivatives of Blaschke products,” Math. Scand., 28, 189–201 (1971).
A. B. Aleksandrov, “Operator Lipschitz functions and linear fractional transformations,” Zap. Nauchn. Semin. POMI, 401, 5–52 (2012).
A. B. Aleksandrov and V. V. Peller, “Operator Hölder–Zygmund functions,” Advances in Math., 224, 910–966 (2010).
A. B. Aleksandrov and V. V. Peller, “Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities,” Indiana Univ. Math. J., 59:4, 1451–1490 (2010).
A. B. Aleksandrov and V. V. Peller, “Operator and commutator moduli of continuity for normal operators,” Proc. London Math. Soc., 105:4, 821–851 (2012).
J. Arazy, T. Barton, and Y. Friedman, “Operator differentiable functions,” Int. Equat. Oper. Theory, 13, 462–487 (1990).
K. M. Dyakonov, “Entire functions of exponentsial type and model spaces in H p,” Zap. Nauchn. Semin. LOMI, 190, 81–100 (1991).
K. M. Dyakonov, “Differentiation in star-invariant subspaces. I. Boundedness and compactness,” J. Funct. Anal., 192, 387–409 (2002).
K. M. Dyakonov, “Meromorphic functions and their derivatives: equivalence of norms,” Indiana Univ. Math. J., 57, 1557–1571 (2008).
O. Frostman, “Sur les produits de Blaschke,” Kungl. Fysiogr. Sällsk. Lund Förh., 12, 169–182 (1942).
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York (1981).
H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, 199, Springer (2000).
S. V. Hruščhev and S. A. Vinogradov, “Inner functions and multipliers of Cauchy type integrals,” Ark. Mat., 19, 23–42 (1981).
K. Yosida, Functional Analysis, Springer-Verlag, Berlin-Göttingen-Heidelberg (1965).
B. E. Johnson and J. P. Williams, “The range of a normal derivation,” Pacific J. Math., 58, 105–122 (1975).
H. Kamowitz, “On operators whose spectrum lies on a circle or a line,” Pacific J. Math., 20, 65–68 (1967).
E. Kissin and V. S. Shulman, “On a problem of J. P. Williams,” Proc. Amer. Math. Soc., 130, 3605–3608 (2002).
E. Kissin and V. S. Shulman, “Classes of operator-smooth functions. I. Operator-Lipschitz functions,” Proc. Edinb. Math. Soc. (2), 48, 151–173 (2005).
E. Kissin and V. S. Shulman, “On fully operator Lipschitz functions,” J. Funct. Anal., 253, 711–728 (2007).
E. Kissin, V. S. Shulman, and L. B. Turowska, “Extension of operator Lipschitz and commutator bounded functions,” Oper. Theory Adv. Appl., 171, 225–244 (2006).
P. Koosis, Introduction to H p Spaces, Cambridge Univ. Press (1980).
K. Kuratowski, Topology, Vol. 1, Academic Press, New York–London; PWN-Polish Scientific Publishers, Warszawa, XX (1966).
I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Moscow (1974).
V. V. Peller, “Hankel operators in the theory of perturbations of unitary and self-adjoint operators,” Funkts. Anal. Prilozh., 19:2, 37–51 (1985).
V. V. Peller, “Hankel operators in the perturbation theory of unbounded self-adjoint operators,” in: Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York (1990), pp. 529–544.
G. Pisier, “Similarity problems and completely bounded maps,” Second, expanded edition. Includes the solution to “The Halmos problem,” Lect. Notes Math., 1618 (2001).
W. Rudin, Function Theory in the Unit Ball of ℂn, Grundlehren der math. Wiss., Bd. 241, Springer-Verlag, New York, Heidelberg, Berlin (1980).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 416, 2013, pp. 5–58.
Rights and permissions
About this article
Cite this article
Aleksandrov, A.B. Operator Lipschitz Functions and Model Spaces. J Math Sci 202, 485–518 (2014). https://doi.org/10.1007/s10958-014-2057-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-2057-y