Operator Lipschitz functions and model spaces

A. B. Aleksandrov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 5-58

Résumé

Let $H^\infty$ denote the space of bounded analytic functions on the upper half plane $\mathbb C_+$. We prove that each function in the model space $H^\infty\cap\Theta\overline{H^\infty}$ is an operator Lipschitz function on $\mathbb R$ if and only if the inner function $\Theta$ is a usual Lipschitz function, i.e., $\Theta'\in H^\infty$. Let $(\mathrm{OL})'(\mathbb R)$ denote the set of all functions $f\in L^\infty$ whose antiderivative is operator Lipschitz on the real line $\mathbb R$. We prove that $H^\infty\cap\Theta\overline{H^\infty}\subset(\mathrm{OL})'(\mathbb R)$ if $\Theta$ is a Blaschke product with the zeros satisfying the uniform Frostman condition. We deal also with the following questions. When does an inner function $\Theta$ belong to $(\mathrm{OL})'(\mathbb R)$? When does each divisor of an inner function $\Theta$ belong to $(\mathrm{OL})'(\mathbb R)$? As an application, we deduce that $(\mathrm{OL})'(\mathbb R)$ is not a subalgebra of $L^\infty(\mathbb R)$. Another application is related to a description of the sets of discontinuity points for the derivatives of the operator Lipschitz functions. We prove that a set $\mathcal E$, $\mathcal E\subset\mathbb R$, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if $\mathcal E$ is an $F_\sigma$ set of first category. A considerable proportion of the results of the paper are based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton and Friedman. We give also a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition.

Bibliographie

[1] P. R. Ahern, D. N. Clark, “Radial Nth derivatives of Blaschke products”, Math. Scand., 28 (1971), 189–201 | MR | Zbl

[2] A. B. Aleksandrov, “Operatorno lipshitsevy funktsii i drobno-lineinye preobrazovaniya”, Zap. nauchn. sem. LOMI, 401, 2012, 5–52 | MR

[3] A. B. Aleksandrov, V. V. Peller, “Operator Hölder–Zygmund functions”, Advances in Math., 224 (2010), 910–966 | DOI | MR | Zbl

[4] A. B. Aleksandrov, V. V. Peller, “Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities”, Indiana Univ. Math. J., 59:4 (2010), 1451–1490 | DOI | MR | Zbl

[5] A. B. Aleksandrov, V. V. Peller, “Operator and commutator moduli of continuity for normal operators”, Proc. London Math. Soc., 105:4 (2012), 821–851 | DOI | MR | Zbl

[6] J. Arazy, T. Barton, Y. Friedman, “Operator differentiable functions”, Int. Equat. Oper. Theory, 13 (1990), 462–487 | MR

[7] K. M. Dyakonov, “Tselye funktsii eksponentsialnogo tipa i modelnye podprostranstva v $H^p$”, Zap. nauchn. sem. LOMI, 190, 1991, 81–100 | MR | Zbl

[8] K. M. Dyakonov, “Differentiation in star-invariant subspaces. I. Boundedness and compactness”, J. Funct. Anal., 192 (2002), 387–409 | DOI | MR | Zbl

[9] K. M. Dyakonov, “Meromorphic functions and their derivatives: equivalence of norms”, Indiana Univ. Math. J., 57:4 (2008), 1557–1571 | DOI | MR | Zbl

[10] O. Frostman, “Sur les produits de Blaschke”, Kungl. Fysiogr. Sällsk. Lund Förh., 12:15 (1942), 169–182 | MR | Zbl

[11] Dzh. Garnett, Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR

[12] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, 199, Springer, 2000 | DOI | MR | Zbl

[13] S. V. Hruščhev, S. A. Vinogradov, “Inner functions and multipliers of Cauchy type integrals”, Ark. Mat., 19 (1981), 23–42 | DOI | MR

[14] K. Iosida, Funktsionalnyi analiz, Mir, M., 1967 | MR

[15] B. E. Johnson, J. P. Williams, “The range of a normal derivation”, Pacific J. Math., 58 (1975), 105–122 | DOI | MR | Zbl

[16] H. Kamowitz, “On operators whose spectrum lies on a circle or a line”, Pacific J. Math., 20 (1967), 65–68 | DOI | MR | Zbl

[17] E. Kissin, V. S. Shulman, “On a problem of J. P. Williams”, Proc. Amer. Math. Soc., 130 (2002), 3605–3608 | DOI | MR | Zbl

[18] E. Kissin, V. S. Shulman, “Classes of operator-smooth functions. I. Operator-Lipschitz functions”, Proc. Edinb. Math. Soc. (2), 48 (2005), 151–173 | DOI | MR | Zbl

[19] E. Kissin, V. S. Shulman, “On fully operator Lipschitz functions”, J. Funct. Anal., 253 (2007), 711–728 | DOI | MR | Zbl

[20] E. Kissin, V. S. Shulman, L. B. Turowska, “Extension of operator Lipschitz and commutator bounded functions. The extended field of operator theory”, Oper. Theory Adv. Appl., 171, Birkhäuser, Basel, 2007, 225–244 | DOI | MR

[21] P. Kusis, Vvedenie v teoriyu prostranstv $H_p$, Mir, M., 1984 | MR

[22] K. Kuratovskii, Topologiya, v. 1, Mir, M., 1966 | MR

[23] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[24] V. V. Peller, “Operatory Gankelya v teorii vozmuschenii unitarnykh i samosopryazhennykh operatorov”, Funkts. analiz i ego pril., 19:2 (1985), 37–51 | MR | Zbl

[25] V. V. Peller, “Hankel operators in the perturbation theory of unbounded self-adjoint operators”, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York, 1990, 529–544 | MR

[26] G. Pisier, Similarity problems and completely bounded maps, Includes the solution to “The Halmos problem”, Lecture Notes in Mathematics, 1618, Second, expanded edition, Springer-Verlag, Berlin, 2001 | DOI | MR | Zbl

[27] U. Rudin, Teoriya funktsii v edinichnom share iz $\mathbb C^n$, Mir, M., 1984 | MR | Zbl