Maps on Complex Manifolds into Grassmann Spaces Defined by Reproducing Kernels of Bergman Type

Monastyrski, Marcin; Pasternak-Winiarski, Zbigniew

doi:10.1515/dema-1997-0225

Cited by 6 publications

(9 citation statements)

References 5 publications

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“…There is a huge literature on different notions of coherent states both in physics and mathematics; see, for instance, [9,13,30,40,42]. Here, the approach suggested in [36] and [37] is taken as a leading motivation.…”

Section: Coherent Statesmentioning

confidence: 99%

“…Maps like K are significant in physics, algebraic geometry, and complex analysis [9,30,37]. With motivation in the physical interpretation given in [36] and [37], any smooth mapping W Z !…”

Section: Theorem 3 ([5 Th 51]) Let … and K Be As Above Letmentioning

confidence: 99%

“…Also, the corresponding reproducing kernel Hilbert spaces are related by H K D .H K / . By using a suitable method of localization of RK Hilbert spaces on vector bundles, one can obtain infinite-dimensional versions of the properties of Bergman kernels established in [30].…”

Section: Theorem 3 ([5 Th 51]) Let … and K Be As Above Letmentioning

confidence: 99%

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Geometric Perspectives on Reproducing Kernels

Beltiţă¹,

Galé²

2015

Operator Theory

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Section: Coherent Statesmentioning

confidence: 99%

“…Maps like K are significant in physics, algebraic geometry, and complex analysis [9,30,37]. With motivation in the physical interpretation given in [36] and [37], any smooth mapping W Z !…”

Section: Theorem 3 ([5 Th 51]) Let … and K Be As Above Letmentioning

confidence: 99%

See 1 more Smart Citation

Geometric Perspectives on Reproducing Kernels

Beltiţă¹,

Galé²

2015

Operator Theory

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“…Quantization maps and kernels. Motivated by the significant physical interpretation given in [Od88] and [Od92] (see also [MP97] and [BG11]) to maps from manifolds into the projective space of a complex Hilbert space, we use the following terminology.…”

Section: Reproducing Kernels and Their Classifying Morphismsmentioning

confidence: 99%

Linear connections for reproducing kernels on vector bundles

Beltiţă

Galé

2013

Math. Z.

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Abstract. We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.

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“…The construction of the mapping ζ K : Z → Gr(H K ) in Theorem 5.1 is inspired by the mapping Z defined in formula (16) in [MP97], which in turn extends the corresponding maps for complex line bundles given in [Od92]. In the latter reference it is shown that categories formed by objects like ζ K (or Z) are equivalent to categories of vector bundles with distinguished kernels.…”

Section: The Universality Theoremsmentioning

confidence: 99%

Universal objects in categories of reproducing kernels

Beltiţă¹,

Galé²

2011

Rev. Mat. Iberoam.

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We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of C * -algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing (− * )-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing (− * )-kernels with universality properties with respect to the operation of pull-back. We show how completely positive maps can be regarded as pull-backs of universal ones linked to the tautological bundle over the Grassmann manifold of the Hilbert space 2 (N).

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Maps on Complex Manifolds into Grassmann Spaces Defined by Reproducing Kernels of Bergman Type

Cited by 6 publications

References 5 publications

Geometric Perspectives on Reproducing Kernels

Geometric Perspectives on Reproducing Kernels

Linear connections for reproducing kernels on vector bundles

Universal objects in categories of reproducing kernels

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