IntroductionThis paper is an attempt to start a detailed analysis of isoperimetric eigenvalue problems in algebras. This kind of problem arises in such fields as buckling of columns, [ 161, [ 171, nuclear reactor control and design, [ 1 31, [ 13'1, [ 141, [3 11, asymptotic distribution of eigenvalues and stability of solutions of periodic ordinary differential equations, [ 191, and others, [29]. The conjectures of Shapiro [27], and Mordell [22], [23] concerning the inequality for xi 2 0, + xi+2 > 0, i = 1, 2, * -(mod n), can be recast in the form of an isoperimetric eigenvalue problem.Let us explain what we mean by an isoperimetric eigenvalue problem and what our results are, avoiding excessive details here. Let A be a linear operator acting on a real or complex commutative algebra B, which has an involution. Consider the eigenvalue equationwhere I is a scalar, x is restricted to a subset X of B and p is restricted to a subset R of B, the elements of which satisfy the normalization condition where f is a positive linear functional on B. consists then in Jinding p so that, say, 2 8 A is extrernized.
The isoperimetric eigenvalue problemThe relevance of this problem becomes clear when B has an identity element e E R ; for then, the eigenvalue equation Ax = I x becomes imbedded in (1).* The research for this paper was partly carried out at the Courant Institute of Mathematical Sciences and was partly sponsored by Nonr Contract NOO14-67-A-0112-0015. Reproduction in whole or in part is permitted for any purpose of the United States Government.
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P. NOWOSADAssume that it is so and that further X c s1, where Q is the set of regular elements of B. Then we get, using (1) and (2), ( 3 )I n this way this functional arises naturally and, in order to be complete, an analysis of our problem must, therefore, necessarily consider the study of such functionals. I n addition, it appears in the inequality described previously, though not in a n immediate form. Let us point out here that a very well-known instance of this functional is given by the line integralwhere g is a meromorphic function and I ' a closed Jordan curve in the complex plane, which does not pass through the zeros and poles ofg.. ) It is interesting that, in a n appropriate sense, the remarkable properties of (4) go over to the general case ( 3 ) under very broad conditions, as will be explained in the sequel. I n order to express this we need some definitions. Given two elements u and v in Q, let w = u-lv and set P(w) = set of polynomials in w and w-l. The set u . P ( w ) contains both u and v and is a module over the algebra P ( w ) ; we call it the module M generated by u and v.Call N the subset of B of elements x such that f ( x * x ) = 0, * being the involution. For convenience we take B to be a *-Banach algebra (some results could be obtained under weaker conditions). Then BIN is a pre-Hilbert space whose completion we denote by Hf. This shows that, as a consequence of the existence of linearly independent points of local minimum, I and A must have a d...