We give a formula for the one-parameter strongly continuous semigroup e −tL , t > 0, generated by the Hermite operator L on the Heisenberg group H 1 in terms of Weyl transforms, and use it to obtain an L 2 estimate for the solution of the initial value problem for the heat equation governed by L in terms of the L p norm of the initial data for 1 ≤ p ≤ ∞.
The Schatten-von Neumann property of a pseudo-differential operator is established by showing that the pseudo-differential operator is a multiplier defined by means of an admissible wavelet associated to a unitary representation of the additive group R" on the C*-algebra of all bounded linear operators from L 2 (R") into L 2 (K"). A bounded linear operator on L 2 (IR) arising in the Landau, Pollak and Slepian model in signal analysis is shown to be a wavelet multiplier studied in this paper.
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