Bergman space
In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions in D for which the p-norm is finite:
The quantity is called the norm of the function f; it is a true norm if . Thus Ap(D) is the subspace of holomorphic functions that are in the space Lp(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:
-
(1)
Thus convergence of a sequence of holomorphic functions in Lp(D) implies also compact convergence, and so the limit function is also holomorphic.
If p = 2, then Ap(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.
Special cases and generalisations
If the domain D is bounded, then the norm is often given by
where is a normalised Lebesgue measure of the complex plane, i.e. dA = dz/Area(D)</. Alternatively dA = dz/π is used, regardless of the area of D. The Bergman space is usually defined on the open unit disk of the complex plane, in which case . In the Hilbert space case, given , we have
that is, A2 is isometrically isomorphic to the weighted ℓp(1/(n+1)) space.[1] In particular the polynomials are dense in A2. Similarly, if D = ℂ+, the right (or the upper) complex half-plane, then
where , that is, A2(ℂ+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).[2][3]
The weighted Bergman space Ap(D) is defined in an analogous way,[1] i.e.
provided that w : D → [0, ∞) is chosen in such way, that is a Banach space (or a Hilbert space, if p = 2). In case where , by a weighted Bergman space [4] we mean the space of all analytic functions f such that
and similarly on the right half-plane (i.e. ) we have[5]
and this space is isometrically isomorphic, via the Laplace transform, to the space ,[6][7] where
(here Γ denotes the Gamma function).
Further generalisations are sometimes considered, for example denotes a weighted Bergman space (often called a Zen space[3]) with respect to a translation-invariant positive regular Borel measure on the closed right complex half-plane , that is
Reproducing kernels
The reproducing kernel of A2 at point is given by[1]
and similarly for we have[5]
- .
In general, if maps a domain conformally onto a domain , then[1]
In weighted case we have[4]
and[5]
References
- 1 2 3 4 Duren, Peter L.; Schuster, Alexander (2004), Bergman spaces, Mathematical Series and Monographs, American Mathematical Society, ISBN 978-0-8218-0810-8
- ↑ Duren, Peter L. (1969), Extension of a theorem of Carleson (PDF), 75, Bulletin of the American Mathematical Society, pp. 143–146
- 1 2 Jacob, Brigit; Partington, Jonathan R.; Pott, Sandra (2013-02-01), On Laplace-Carleson embedding theorems, 264 (3), Journal of Functional Analysis, pp. 783–814
- 1 2 Cowen, Carl; MacCluer, Barbara (1995-04-27), Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, p. 27, ISBN 9780849384929
- 1 2 3 Elliott, Sam J.; Wynn, Andrew (2011), Composition Operators on the Weighted Bergman Spaces of the Half-Plane, 54 (2), Proceedings of the Edinburgh Mathematical Society, pp. 374–379
- ↑ Duren, Peter L.; Gallardo-Gutiérez, Eva A.; Montes-Rodríguez, Alfonso (2007-06-03), A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces (PDF), 39 (3), Bulletin of the London Mathematical Society, pp. 459–466
- ↑ Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts (PDF), 62 (1), Journal of Operator Theory, pp. 199–214
Further reading
- Bergman, Stefan (1970), The kernel function and conformal mapping, Mathematical Surveys, 5 (2nd ed.), American Mathematical Society
- Hedenmalm, H.; Korenblum, B.; Zhu, K. (2000), Theory of Bergman Spaces, Springer, ISBN 978-0-387-98791-0
- Richter, Stefan (2001), "Bergman spaces", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
See also
- Bergman kernel
- Banach space
- Hilbert space
- Reproducing kernel Hilbert space
- Hardy space
- Dirichlet space