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| 2. |
- Ben Said, S., et al.
(author)
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Invariant trilinear forms on spherical principal series of real rank one semisimple Lie groups
- 2014
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In: International Journal of Mathematics. - : World Scientific Pub Co Pte Lt. - 0129-167X .- 1793-6519. ; 25:3
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Journal article (peer-reviewed)abstract
- Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form K. indexed by a complex multiparameter (v)under bar = (v1, v2, v3) and defined on the space of smooth functions on the product of three spheres in F-n, where F is either R, C, H, or O with n = 2. We then study the analytic continuation of the trilinear form with respect to (v1, v2, v3), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called "partial intertwining operator", we obtain an expression for the generalized Bernstein-Reznikov integral K-(v) under bar (1 circle times 1 circle times 1) in terms of hypergeometric functions.
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| 3. |
- Englis, M., et al.
(author)
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CONNECTION AND CURVATURE ON BUNDLES OF BERGMAN AND HARDY SPACES
- 2020
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In: Documenta Mathematica. - 1431-0643 .- 1431-0635. ; 25, s. 189-217
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Journal article (peer-reviewed)abstract
- We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(z, w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation del(z) on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R-(k)(Z,Z) for large k and for the induced connection [del((k))(Z), T-f((k))] on Toeplitz operators T-f. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.
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| 4. |
- Englis, M., et al.
(author)
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Hankel Operators and the Dixmier Trace on Strictly Pseudoconvex Domains
- 2010
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In: Documenta Mathematica. - 1431-0643 .- 1431-0635. ; 15, s. 601-622
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Journal article (peer-reviewed)abstract
- Generalizing earlier results for the disc and the ball, we give a formula for the Dixmier trace of the product of 2(n) Hankel operators on Bergman spaces of strictly pseudoconvex domains in C-n. The answer turns out to involve the dual Levi form evaluated on boundary derivatives of the symbols. Our main tool is the theory of generalized Toeplitz operators due to Boutet de Monvel and Guillemin.
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| 5. |
- Englis, M., et al.
(author)
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Hankel operators and the Dixmier trace on the Hardy space
- 2016
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In: Journal of the London Mathematical Society. - : Wiley. - 0024-6107 .- 1469-7750. ; 94, s. 337-356
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Journal article (peer-reviewed)abstract
- We give criteria for the membership of Hankel operators on the Hardy space on the disc in the Dixmier class, and establish estimates for their Dixmier trace. In contrast to the situation in the Bergman space setting, it turns out that there exist Dixmier-class Hankel operators that are not measurable (that is, their Dixmier trace depends on the choice of the underlying Banach limit), as well as Dixmier-class Hankel operators that do not belong to the Schatten-Lorentz ideal. A related question concerning logarithmic interpolation of Besov spaces is also discussed.
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| 7. |
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| 8. |
- Englis, M., et al.
(author)
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Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces
- 2010
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In: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 0025-5874 .- 1432-1823. ; 264:4, s. 901-912
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Journal article (peer-reviewed)abstract
- We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb CnCn for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szegö kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.
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| 9. |
- Englis, M., et al.
(author)
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Toeplitz and Hankel operators and Dixmier traces on the unit ball of Cn
- 2009
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In: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 137:11, s. 3669-3678
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Journal article (peer-reviewed)abstract
- We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of ℂd. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.
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| 10. |
- Englis, M., et al.
(author)
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Toeplitz Operators on Higher Cauchy-Riemann Spaces
- 2017
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In: Documenta Mathematica. - 1431-0643 .- 1431-0635. ; 22, s. 1081-1116
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Journal article (peer-reviewed)abstract
- We develop a theory of Toeplitz, and to some extent Hankel, operators on the kernels of powers of the boundary d-bar operator, suggested by Boutet de Monvel and Guillemin, and on their analogues, somewhat better from the point of view of complex analysis, defined using instead the covariant Cauchy-Riemann operators of Peetre and the second author. For the former, Dixmier class membership of these Hankel operators is also discussed. Our main tool are the generalized Toeplitz operators (with pseudodifferential symbols), in particular there appears naturally the problem of finding parametrices of matrices of such operators in situations when the principal symbol fails to be elliptic.
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