On the Spectral Theory of Singular Integral Operators.

THEOREM 4. Let a be an arbitrarily often differentiable function on the real line having countably many isolated zeros s1, S2, . . . offinite orders ml, M2, ..., respectively. Denote by A the multiplication operator mapping x into Ax = ax and let E be any linear space satisfying X = E (3 R*. Then there is a unique continuous linear op rator A-l :X -X whose range is orthogonal to E and is such that AA-1 = I. The boundedness and local property of PE can be proved by using the fact that every E satisfying E n R* = {a} is locally finite dimensional. Precisely, X = E @ R* if and only if for every root Sk there exist elements ex e E (X = 0, ... Mk1) with the property that EX(M)(5k) = 6d,, for X, ,u = 0, . . ., mk 1, where 6),,5 is the Kronecker delta and at each s,, (n + k) the functions Ex vanish with order Mn. A natural division operator (A-1)*:R* X is defined on R*, which maps r* eR* into r*/a E X. Although the operator (A-')* is not continuous, it can be extended in many ways to a linear operator (A -1) *:X -X. Namely, each A-1 haan adj oint (A -1) *:X -* X whose restriction to R * is the natural division operator. The null-space of the adj oint operator associated with E is the linear space E. T>.e crucial point in the proof is the verification of the condition "[u:au E W] I) 1 ,'s to Au for every W in 'U." The only non-trivial tool needed in the proof of this p. po-ition is the generalized mean-value theorem of the differential calculus. The Dame proposition leads to the solution of the division problem for special types of divisors in the case of several variables. This includes division by linear functions and by quadratic functions of the form a(si, ... Sn) = Eaksk2, where ak . o (k = 1, . .., n). Earlier Schwartz proved that division of individual distributions by "regular" functions is possible.' The principle of localization and a change in variables show that division by regular functions can be treated globally: THEOREM 5. Let a be a regular function on a finite dimensional Euclidean space S and let A :X -X be the operator corresponding to multiplication by a. Then there exist continuous linear operators A-1 :X X such that AA -1 = I. A satisfactory solution of the division problem in general would depend on the proof of the proposition " [u:au E W] belongs to A for every W in A."