Hankel operators and Plancherel formula
Abstract: We consider the natural unitary action of the group SU(d, 1) on the Hilbert space L2(B,d/j,\), where B is the unit ball of C , dfi\ is the measure K(z, z)~~'+ï dm(z), and K(z, z) the reproducing kernel of the Bergman space on B. For A > —1 and not an integer, a Plancherel formula is proved for this action and, in particular, an explicit formula for the correponding Plancherel measure is found. The group actions on discrete submodules appearing in the Plancherel formula are isomorphic to actions on certain holomorphic tensor fields. The discrete parts are alternatively described by using certain invariant Cauchy-Riemann operators. We also find some bases in the discrete parts. In the case of the unit disk, orthogonal bases are obtained, which involves certain orthogonal polynomials introduced by Romanovski in statistics.A similar problem is studied for the spaces of Hilbert-Schmidt operators (forms) on weighted Bergman spaces in the case of the unit disk. The corresponding Plancherel formula is obtained and explicit realizations of the discrete parts are found. The operators in the discrete parts are certain Hankel or Toeplitz type operators. Their Schatten-von Neumann properties are studied. This completes earlier work by Janson and Peetre.We also study the Hankel forms on an annulus and establish their Schatten von Neumann properties.
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