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Zbl 0752.22002
Ali, S.T.; Antoine, J.-P.; Gazeau, J.-P.
Square integrability of group representations on homogeneous spaces. I: Reproducing triples and frames. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 55, No.4, 829-855 (1991). ISSN 0246-0211

Given a Hilbert space $H$, a regular positive operator (on $H$) valued measure $\mu$ on a locally compact space $X$ such that $\mu(X)=A\in{\cal L}(H)\sp +$, a positive regular Borel measure $\nu$ on $X$, a weakly measurable positive operator (on $H$) valued function $F$ on $X$ such that $\mu(\Delta)=\int\sb \Delta F(x)d\nu(x)$ $(\Delta$ Borel set of $X)$, the triple $(H,F,A)$ is called reproducing triple. Such an object generates an overcomplete family of vectors in $H$ which has most of the attributes of the familiar coherent states. On the other hand, given a reproducing triple, one can associate to it a unitarily equivalent one whose Hilbert space carries a reproducing kernel. In the particular case where $F$ is a constant function with finite rank operator values, and $A\sp{-1}$ exists and is bounded, the notion of reproducing triple generalises the one of frame in the discrete case. In part II (reviewed below), the notion of reproducing triple will be used to generalize the concept of square integrable representations and their associated coherent states.
[G.Loupias (Montpellier)]

MSC 2000:: *22D12 Other representations of locally compact groups
43A85 Analysis on homogeneous spaces
81R30 Coherent states in quantum theory
22E70 Appl. of Lie groups to physics

Keywords: regular positive operator valued measure; locally compact space; reproducing triple; coherent states; reproducing kernel; frame

Citations: Zbl 0752.22003

Cited in: Zbl 1083.42032 Zbl 1053.81046 Zbl 0874.22004 Zbl 0752.22003

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