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By Robert C. Gunning, Hugo Rossi

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Non-negative symbols yield nonnegative operators. This is not true for the Weyl quantization, but the result for Anti-Wick operators can be fruitfully used to prove lower bounds for Weyl operators. , satisfying for every α, β ∈ Nd , |∂ξα ∂xβ a(x, ξ)| ≤ Cαβ , (x, ξ) ∈ R2d . The functions Gy,η can in fact be used to almost diagonalize such operators, in the sense that, after conjugating with the short-time Fourier transform, the operator V a(x, D)V ∗ has an integral kernel K(y , η ; y, η) dominated by a convolution kernel in L1 .

First we present the needed results from the abstract theory of Fredholm operators in Hilbert spaces. 46 Chapter 1. 1 Abstract Theory Let H1 , H2 be complex Hilbert spaces and B(H1 , H2 ) be the space of bounded operators H1 → H2 . We define the kernel and cokernel of an operator T ∈ B(H1 , H2 ) respectively as Ker T = {x ∈ H1 : T x = 0}, Coker T = H2 /T (H1 ). Since T is bounded, Ker T is a closed subspace of H1 . However, T (H1 ) need not be closed. , W ∩ T (H1 ) = {0} and W + T (H1 ) = H2 , then W is isomorphic to H2 /T (H1 ) via the map x → x + T (H1 ).

Of course, it suffices to verify that Au L2 (Rd ) + Ru L2 (Rd ) Au L2 (Rd ) + Ru L2 (Rd ) , u ∈ S(Rd ). 3). 1. Since R and RB are regularizing operators, and therefore bounded on L2 (Rd ), we obtain Au L2 (Rd ) + Ru L2 (Rd ) Au L2 (Rd ) + Ru L2 (Rd ) + AR u L2 (Rd ) . If we replace again u = B A u − R u in the last term of this expression and argue similarly, we obtain the desired estimate. (b) The only non-trivial fact is the completeness of H(M ). To prove it, let un be a Cauchy sequence in H(M ).

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