On the Friedrichs operator
HTML articles powered by AMS MathViewer
- by Peng Lin and Richard Rochberg PDF
- Proc. Amer. Math. Soc. 123 (1995), 3335-3342 Request permission
Abstract:
Let $\Omega$ be a simply connected domain in ${\mathbb {C}^1}$ with the area measure dA. Let ${\bar P_\Omega }$ be the orthogonal projection from ${L^2}(\Omega ,dA)$ onto the closed subspace of antiholomorphic functions in ${L^2}(\Omega ,dA)$. The Friedrichs operator ${\bar T_\Omega }$ associated to $\Omega$ is the operator from the Bergman space $L_a^2(\Omega )$ into ${L^2}(\Omega ,dA)$ defined by ${\bar T_\Omega }f = {\bar P_\Omega }f$. In this note, some smoothness conditions on the boundary of $\Omega$ are given such that the Friedrichs operator ${\bar T_\Omega }$ belongs to the Schatten classes ${S_p}$.References
- Stefan Bergman, The Kernel Function and Conformal Mapping, Mathematical Surveys, No. 5, American Mathematical Society, New York, N. Y., 1950. MR 0038439, DOI 10.1090/surv/005
- Frank Beatrous and Jacob Burbea, Holomorphic Sobolev spaces on the ball, Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60. MR 1010151
- Kurt Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41 (1937), no. 3, 321–364. MR 1501907, DOI 10.1090/S0002-9947-1937-1501907-0
- Daniel H. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk, J. Funct. Anal. 110 (1992), no. 2, 247–271. MR 1194989, DOI 10.1016/0022-1236(92)90034-G S. Norman, The Friedrichs operator, research report, TRITAMAT, Royal Inst. of Technology 6 (1987).
- Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
- Harold S. Shapiro, Domains allowing exact quadrature identities for harmonic functions—an approach based on p.d.e, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 335–354. MR 820535 —, The Schwarz function and its generalization to higher dimensions, Wiley, New York, 1993.
- Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990. MR 1074007
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3335-3342
- MSC: Primary 47B38; Secondary 32A37, 32H10, 46E99, 47B10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1264822-8
- MathSciNet review: 1264822