By Edgar Asplund; Lutz Bungart

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**Extra resources for A first course in integration**

**Example text**

14 below. 6 we can limit ourselves to considering the standard (left) quantization. Here we regard a(x, D) as a continuous map S (Rd ) → S (Rd ) (cf. 13). The boundedness of a(x, D) on L2 (Rd ) is then easily seen to be equivalent to the estimate a(x, D)u u , u ∈ S(Rd ), where u = u L2 (Rd ) . Indeed, since S(Rd ) is dense in L2 (Rd ), this estimate tells us that the restriction of a(x, D) to S(Rd ) (which maps S(Rd ) into itself) extends to a bounded operator on L2 (Rd ). This extension coincides with the restriction of a(x, D) to L2 (Rd ) by the uniqueness of the limit in S (Rd ).

5. Let M, M1 , M2 , M be regular weights and let P be a pseudodiﬀerential operator with symbol in S(M ; Φ, Ψ). (a) P deﬁnes a bounded operator from H(M ) into H(M/M ). (b) If moreover M (x, ξ)M2 (x, ξ)/M1 (x, ξ) tends to zero at inﬁnity, P deﬁnes a compact operator H(M1 ) → H(M2 ). Proof. (a) Let A, B, R be as at the beginning of this section, so that the norm in H(M ) is deﬁned in terms of A and R. Let A be a pseudo-diﬀerential operator with elliptic symbol in S(M/M ), with parametrix B having symbol in S(M /M ), and R = B A − I.

3. Let T ∈ Fred(H1 , H2 ), and S ∈ B(H1 , H2 ) with S suﬃciently small. Then dim Ker(T + S) ≤ dim Ker T , T + S is Fredholm and ind(T + S) = ind T . Proof. Let V ⊂ H1 be any closed subspace of ﬁnite codimension, such that V ∩ Ker T = {0}. Since T (V ) has ﬁnite codimension in T (H2 ), which has ﬁnite codimension in H2 , we see that T (V ) has ﬁnite codimension in H2 . 2. Moreover, the operator T induces an operator T˜ : H1 /V → H2 /T (V ), acting therefore between ﬁnite dimensional spaces. A direct inspection shows that the natural map Ker T → H1 → H1 /V deﬁnes an isomorphism Ker T → Ker T˜, whereas the natural map H2 /T (H1 ) → H2 /T (V ) → (H2 /T (V )) /T˜(H1 /V ) deﬁnes an isomorphism Coker T → Coker T˜.

### A first course in integration by Edgar Asplund; Lutz Bungart

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