×

Trace ideal criteria for Toeplitz operators. (English) Zbl 0618.47018

For a complex measure \(\mu\) on the open unit disk U define an operator \(T_{\mu}\) on a Hilbert space H of analytic functions with reproducing kernel k(z,w) by \(T_{\mu}f(w)=\int f(z)\overline{k(z,w)}d\mu (z)\). For a certain scale of Hilbert spaces \(H_{\alpha}\), \(\alpha <1\), which includes the Hardy space \(H^ 2\) and weighted Bergman spaces \(A^{2,\beta}\), conditions are obtained which imply \(T_{\mu}\) belongs to a Schatten ideal \({\mathcal S}_ p\). If \(\mu\) is a positive measure then these conditions are necessary and sufficient. Application to composition operators and restriction operators are indicated.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Amar, E, Suites d’interpolation pour LES classes de Bergman de la boule et du polydisque de \(C\)^{n}, Canada J. math., 30, 711-737, (1978) · Zbl 0385.32014
[2] Carleson, L, Interpolations by bounded analytic functions and the corona problem, Ann. math., 76, 547-559, (1962) · Zbl 0112.29702
[3] Coifman, R; Rochberg, R, Representation theorems for holomorphic and harmonic functions in Lp, Astérisque, 77, 11-66, (1980) · Zbl 0472.46040
[4] Gohberg, I.C; Krein, M.G; Gohberg, I.C; Krein, M.G, Introduction to the theory of linear non-selfadjoint operators, (), (1965), Nauka Moscow, English translation · Zbl 0106.08905
[5] Gohberg, I.C; Markus, A.S; Gohberg, I.C; Markus, A.S, Some relations between eigenvalues and matric elements of linear operators, Mat. sb., Amer. math. soc. transl., 52, 106, 201-216, (1966), (2)
[6] Luecking, D.H, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. math., 107, 85-111, (1985) · Zbl 0584.46042
[7] Luecking, D.H, Representation and duality in weighted spaces of analytic functions, Indiana univ. math. J., 34, 319-336, (1985) · Zbl 0538.32004
[8] MacCluer, B.D; Shapiro, J.H, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Can. J. math., 38, 878-906, (1986) · Zbl 0608.30050
[9] McDonald, G; Sundberg, C, Toeplitz operators on the disk, Indiana univ. math. J., 28, 595-611, (1979) · Zbl 0439.47022
[10] Rochberg, R, Trace ideal criteria for Hankel operators and commutators, Indiana univ. math. J., 31, 913-925, (1982) · Zbl 0514.47020
[11] Rudin, W, Function theory on the unit ball of \(C\)^{n}, (1980), Springer-Verlag New York
[12] Semmes, S, Trace ideal criteria for Hankel operators and application to Besov spaces, Integral equations oper. theory, 7, 241-281, (1984) · Zbl 0541.47023
[13] Simon, B, Trace ideals and their applications, () · Zbl 1074.47001
[14] Stegenga, D, Multipliers of the Dirichlet space, Illinois J. math., 24, 113-139, (1980) · Zbl 0432.30016
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.