zbMATH — the first resource for mathematics

Differentiation in star-invariant subspaces. II: Schatten class criteria. (English) Zbl 1011.47006
Given an inner function \(\theta\) on the upper half-plane, let the space \(K_\theta: =H^2\cap \theta\overline {H^2}\) be such that the operator \({d\over dy}: K_\theta\to L^2\) is compact. As follows from the results obtained in first part of the paper [ibid. 364-386 (2002; Zbl 1011.47005)], then \(\theta\) is necessarily a Blaschke product \(B\) whose zeros \(z_j\) satisfy \(\text{Im} z_j\to\infty\), and the operator \({d\over dx}\) is henceforth regarded as acting from \(K_B\) to \(K_{B^2}\). The problem under consideration is to determine when the operator belongs to the Schatten-von Neumann classes \(s_p\) (i.e., if its \(s\)-numbers belong to \(l^p)\). A complete solution is given for \(p=1\) and \(p=2\) (the cases of nuclear and Hilbert-Schmidt operators), and explicit formulas for the trace and the Hilbert-Schmidt norm are presented. For other values of \(p\), some necessary and some sufficient conditions are found.

47A15 Invariant subspaces of linear operators
30D50 Blaschke products, etc. (MSC2000)
30D55 \(H^p\)-classes (MSC2000)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
Full Text: DOI
[1] M. S. Birman, and, M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Mathematics and its Applications, Reidel, Dordrecht, 1987.
[2] Clark, D.N., One-dimensional perturbations of restricted shifts, J. anal. math., 25, 169-191, (1972) · Zbl 0252.47010
[3] K. M. Dyakonov, Entire functions of exponential type and model subspaces in, H^p, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov, 190, 1991, 81, - 100, (English transl. in, J. Math. Sci., \bf71, 1994, 2222-2233).
[4] Dyakonov, K.M., On the zeros and Fourier transforms of entire functions in the paley – wiener space, Math. proc. Cambridge philos. soc., 119, 357-362, (1996) · Zbl 0851.30013
[5] Dyakonov, K.M., Factorization of smooth analytic functions via hilbert – schmidt operators, Saint |St. Petersburg math. J., 8, 543-569, (1997)
[6] Dyakonov, K.M., Differentiation in star-invariant subspaces I: boundedness and compactness, J. funct. anal., 192, 364-386, (2002) · Zbl 1011.47005
[7] Garnett, J.B., Bounded analytic functions, (1981), Academic Press New York · Zbl 0469.30024
[8] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1952), Cambridge Univ. Press Cambridge
[9] Nikol’skiĭ, N.K., Treatise on the shift operator, (1986), Springer-Verlag Berlin
[10] Russo, B., On the hausdorff – young theorem for integral operators, Pacific J. math., 68, 241-253, (1977) · Zbl 0367.47028
[11] Simon, B., Trace ideals and their applications, (1979), Cambridge Univ. Press Cambridge · Zbl 0423.47001
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.