×

zbMATH — the first resource for mathematics

Hermitian weighted composition operators and Bergman extremal functions. (English) Zbl 1296.47025
For an analytic function \(f\) on the unit disk \(\mathbb D\) and an analytic self-map \(\varphi\) of \(\mathbb D\), let \(W_{f, \varphi}\) be the weighted composition operator defined by \(W_{f, \varphi}h = f (h\circ \varphi)\) for functions \(h\) analytic on \(\mathbb D\). Let \(H^2(\beta)\) be the weighted Hardy space over \(\mathbb D\) with weight sequence \(\beta\). The weighted Hardy space whose reproducing kernel is \((1-z\bar w )^{-\kappa}\), \(\kappa\geq 1\), is denoted by \(H^2(\beta_\kappa)\). So, \(H^2(\beta_\kappa)\) is the Hardy space for \(\kappa=1\) and is the standard weighted Bergman space for \(\kappa>1\).
The authors first observe necessary conditions for \(W_{f, \varphi}\) to be Hermitian and bounded on a given \(H^2(\beta)\). The necessary conditions assert that \(f\) and \(\varphi\) must be linear fractional maps of special type. Those necessary conditions are also shown to be sufficient when the spaces are restricted to \(H^2(\beta_\kappa)\), \(\kappa \geq -1\). Moreover, as in the Hardy space case due to two of the present authors [C. C. Cowen and E. Ko, Trans. Am. Math. Soc. 362, No. 11, 5771–5801 (2010; Zbl 1213.47034)], they show on \(H^2(\beta_\kappa)\) that the Hermitian weighted composition operators are divided into three classes: (i) compact operators, (ii) multiples of isometries, and (iii) those that have no eigenvalues. In the first two cases, the eigenvalues and eigenvectors are explicitly computed, and in the third case additional properties of being cyclic and being part of an analytic semigroup that includes normal weighted composition operators are proved. Finally, for the weighted Bergman spaces \(H^2(\beta_\kappa)\), \(\kappa>1\), the authors apply one of their results in the present paper to compute the extremal functions for the invariant subspaces associated with the usual atomic inner functions \(e^{c {{1+z}\over{1-z}}}\), \(c<0\), and obtain explicit formulas for the projections of the reproducing kernel functions to those invariant subspaces. Those extremal functions and projected reproducing kernel functions are explicitly described in terms of the incomplete Gamma function.

MSC:
47B33 Linear composition operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B38 Linear operators on function spaces (general)
30H20 Bergman spaces and Fock spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bourdon P., Narayan S.: Normal weighted composition operators on the Hardy space $${H\^{2}({\(\backslash\)mathbb U})}$$ J. Math. Anal. Appl. 367, 278–286 (2010) · Zbl 1195.47013 · doi:10.1016/j.jmaa.2010.01.006
[2] Conway J.: A Course in Functional Analysis. Springer, New York (1990) · Zbl 0706.46003
[3] Cowen C.: The commutant of an analytic Toeplitz operator. Trans. Am. Math. Soc. 239, 1–31 (1978) · Zbl 0391.47014 · doi:10.1090/S0002-9947-1978-0482347-9
[4] Cowen C.: An analytic Toeplitz operator that commutes with a compact operator. J. Funct. Anal. 36(2), 169–184 (1980) · Zbl 0438.47029 · doi:10.1016/0022-1236(80)90098-1
[5] Cowen C.: Composition operators on H 2. J. Oper. Theory 9, 77–106 (1983) · Zbl 0504.47032
[6] Cowen C.: Linear fractional composition operators on H 2. Integr. Equ. Oper. Theory 11, 151–160 (1988) · Zbl 0638.47027 · doi:10.1007/BF01272115
[7] Cowen, C., Gallardo-Gutiérrez, E.: The adjoint of a composition operator. Preprint, 1/31/2005
[8] Cowen C., Gallardo-Gutiérrez E.: Projected and multiple valued weighted composition operators. J. Funct. Anal. 238, 447–462 (2006) · Zbl 1106.47023 · doi:10.1016/j.jfa.2006.04.031
[9] Cowen C., Ko E.: Hermitian weighted composition operators on H 2. Trans. Am. Math. Soc. 362, 5771–5801 (2010) · Zbl 1213.47034 · doi:10.1090/S0002-9947-2010-05043-3
[10] Cowen C., MacCluer B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) · Zbl 0873.47017
[11] Cowen C., MacCluer B.: Linear fractional maps of the ball and their composition operators. Acta Sci. Math. (Szeged) 66, 351–376 (2000) · Zbl 0970.47011
[12] Duren P., Schuster A.: Bergman Spaces. American Mathematical Society, Providence (2004)
[13] Forelli F.: The isometries of H p . Can. J. Math. 16, 721–728 (1964) · Zbl 0132.09403 · doi:10.4153/CJM-1964-068-3
[14] Gradshteyn I., Ryzhik I.: Table of Integrals, Series, and Products, 5th edn. Academic Press, San Diego (1994) · Zbl 0918.65002
[15] Gunatillake, G.: Weighted composition operators. Thesis, Purdue University (2005)
[16] Gunatillake G.: The spectrum of a weighted composition operator. Proc. Am. Math. Soc. 135, 461–467 (2007) · Zbl 1112.47019 · doi:10.1090/S0002-9939-06-08497-8
[17] Hedenmalm H., Korenblum B., Zhu K.: Theory of Bergman Spaces. Springer, New York (2000) · Zbl 0955.32003
[18] Hille E., Phillips R.: Functional Analysis and Semigroups, revised edn. American Mathematical Society, Providence (1957) · Zbl 0078.10004
[19] Konig W.: Semicocycles and weighted composition semigroups on H p . Mich. Math. J. 37, 469–476 (1990) · Zbl 0744.47034 · doi:10.1307/mmj/1029004204
[20] Korenblum B.: Cyclic elements in some spaces of analytic functions. Bull. Am. Math. Soc. 5, 317–318 (1981) · Zbl 0473.30031 · doi:10.1090/S0273-0979-1981-14943-0
[21] Kriete T., Moorhouse J.: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359, 2915–2944 (2007) · Zbl 1115.47023 · doi:10.1090/S0002-9947-07-04166-9
[22] Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) · Zbl 0516.47023
[23] Roberts J.: Cyclic inner functions in the Bergman spaces and weak outer functions in H p , 0 &lt; p &lt; 1. Ill. J. Math. 29, 25–38 (1985) · Zbl 0562.30041
[24] Shapiro H., Shields A.: On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Zeit. 80, 217–229 (1962) · Zbl 0115.06301 · doi:10.1007/BF01162379
[25] Siskakis A.: Weighted composition semigroups on Hardy spaces. Linear Alg. Appl. 84, 359–371 (1986) · Zbl 0629.47032 · doi:10.1016/0024-3795(86)90327-7
[26] Siskakis A.: Semigroups of composition operators on spaces of analytic functions, a review. In Studies on Composition Operators. Contemporary Math., 213, 229–252 (1998) · Zbl 0904.47030 · doi:10.1090/conm/213/02862
[27] Yang R., Zhu K.: The root operator on invariant subspaces of the Bergman space. Ill. J. Math. 47, 1227–1242 (2003) · Zbl 1046.47022
[28] Yang, W.: The reproducing kernel of an invariant subspace of the Bergman space. Thesis, SUNY at Albany (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.