By M.W. Wong

The sensible analytic homes of Weyl transforms as bounded linear operators on $ L^{2}({\Bbb R}^{n}) $ are studied when it comes to the symbols of the transforms. The boundedness, the compactness, the spectrum and the sensible calculus of the Weyl remodel are proved intimately. New effects and methods at the boundedness and compactness of the Weyl transforms by way of the symbols in $ L^{r}({\Bbb R}^{2n}) $ and by way of the Wigner transforms of Hermite capabilities are given. the jobs of the Heisenberg workforce and the symplectic team within the learn of the constitution of the Weyl rework are explicated, and the connections of the Weyl remodel with quantization are highlighted during the e-book. Localization operators, first studied as filters in sign research, are proven to be Weyl transforms with symbols expressed when it comes to the admissible wavelets of the localization operators. the implications and techniques during this publication will be of curiosity to graduate scholars and mathematicians operating in Fourier research, operator idea, pseudo-differential operators and mathematical physics. history fabrics are given in sufficient aspect to permit a graduate scholar to continue speedily from the very fundamentals to the frontier of analysis in a space of operator conception.

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Weyl Transforms

The sensible analytic homes of Weyl transforms as bounded linear operators on $ L^{2}({\Bbb R}^{n}) $ are studied by way of the symbols of the transforms. The boundedness, the compactness, the spectrum and the practical calculus of the Weyl remodel are proved intimately. New effects and strategies at the boundedness and compactness of the Weyl transforms by way of the symbols in $ L^{r}({\Bbb R}^{2n}) $ and by way of the Wigner transforms of Hermite capabilities are given.

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Then Wσ : L2 (Rn ) → L2 (Rn ) is a compact operator. 3, we use the following lemma. 4. Let Sh be the Hilbert–Schmidt operator on L2 (Rn ) corresponding to the kernel h in L2 (R2n ). Then Sh is compact. Proof. 11) in L (R ) as k → ∞. 12) as k → ∞. 12), Sh is compact if we can prove that each Stk is compact. To prove that each Stk is compact, it is enough to prove that Sa⊗b is compact for all a and b in L2 (Rn ). But if a and b are in L2 (Rn ), then (Sa⊗b f )(x) Rn a(x)b(y)f (y)dy f, b a(x), x ∈ Rn , for all f in L2 (Rn ).

The answer to the question is yes. 16) for some h in L (R ), then we let σ be the function on R 2 2n n/2 σ (2π ) K −1 h. 19), Wσ Sh A. h. 19) 7 H ∗-Algebras and the Weyl Calculus This chapter is an account, based on the paper [20] by Pool, of a functional calculus for the Weyl transform with symbol in L2 (R2n ). In addition to the identification of Weyl transforms with symbols in L2 (R2n ) with Hilbert–Schmidt operators on L2 (Rn ) proved in the preceding chapter, we need the notion of an H ∗ -algebra studied by Ambrose in [1].

7. 5) for all λ in R and (Rα,β (z, t)f )(x) ei(α·q+β·p) f (x) for all α and β in Rn , x in Rn , f ∈ L2 (Rn ), and (z, t) (q + ip, t) in H n . Thus, the only nontrivial irreducible and unitary representations of H n on L2 (Rn ) are given by {R λ : −∞ < λ < ∞}. 8. 1) for all q and p in Rn , is equal to R 1 (z, 0), where z q + ip. This observation will be useful to us in the next chapter. 9 The Twisted Convolution The aim of this chapter is to express the symbol of the product of two Weyl transforms with symbols in L2 (R2n ) in terms of a twisted convolution, which we now define.

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