By V. E. Nazaĭkinskiĭ, Viktor Evgenʹevich Shatalov, Boris Yu. Sternin
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Extra info for Methods of Noncommutative Analysis: Theory and Applications
Example text
Xn ] is considered. Then it is natural to try to extend the mapping p, 1 „ from C[x i , . . ,An properties 1°-3° being preserved and then, if it is possible, to set 1 f (Ai, •••, n def An) = Ai n (f). ,A n This approach can lead to valuable results if the desired extension not only exists (its existence is merely a restriction on the class of admissible operators A 1 , . . , An ) but is unique. Indeed, under lack of uniqueness the equation A = B apparently would not imply that f (A) = f (B), and the theory would appear to be stillborn.
Its simplicity is just due to the wizardry of our notation, which displays itself in quite a few places thereafter. T2 to F1. F,i ) are infinitely differentiable. We could choose another way and postulate that the operator d „ — : Pi —> Pi dx is continuous. Then the formula 1 Sx 0 dx — t)y)d -c 2. 1 is satisfied. F2 is continuous. 3 Basic Properties Let A be an algebra of operators, and let Fi = . ,-Fn = Y6 n , . . be algebras of symbols. Given a tuple A = (A 1 , . . , An ) of generators in A, we intend to study the mapping n : Fn —> A.
6 Let A, C e A be arbitrary generators; let B, D e A; and suppose that AB = BC ± D. F we have 2 3f 3 1 f (A)B = Bf (C) ± D—(A, C). Sx Proof This can be proved by the following computation: 3 1 sf 3 1 2 1 23 f (A)B — Bf (C) = B(f (A) — f (C)) = B (A — C)—(A, C) Sx 23 1 6f 4 0 = B (A — C)— (A, C) (moving indices apart) Sx 22 3 1 3f 4 0 = 1[B (A — C) — (A, C) (extraction of a linear factor) Sx 2 3f 3 1 = D—(A, A), Sx 0 as desired. 2. This realization deals with symbols of tempered growth and the related class of generators and is particularly needed in applications to differential equations.