Operators Of Odd Order Research Articles

On commuting differential operators of rank 2

We study examples of formally self-adjoint commuting ordinary differential operators of order 4 or 4g + 2 whose coefficients are analytic on ℂ. We prove that these operators do not commute with the operators of odd order, justifying rigorously that these operators are of rank 2.

On Deficiency Indices of Singular Operators of Odd Order

One of the main problems of the spectral theory of linear ordinary differential operators is the study of their deficiency indices depending on the behavior of the coefficients of the corresponding differential expression ly . Such problems were investigated by Naimark [1], Fedoryuk [2], Sultanaev, and others (see the bibliography in [2]). As a rule, symmetric operators with real coefficients were studied. For the case of complex coefficients, Fedoryuk obtained asymptotic formulas for the fundamental system of solutions of the equation ly = 0 under the condition that the coefficients of the differential expression ly are polynomials subject to a rigid constraint on their degree. The paper [3] was devoted to the study of a symmetric differential operator of (2n+1)th order with complex-valued coefficients of the odd powers. Classes of differential operators with deficiency indices (m, m) , where 1 ≤ m ≤ 2n − 1 , were described. The papers [4, 5] were also devoted to the study of symmetric operators in the case of complex coefficients under regularity conditions of Titchmarsh– Levitan type; asymptotic formulas for the fundamental system of solutions of the equation ly = λy were obtained, and in certain special cases the deficiency indices of the corresponding minimum operator were found. It is well known that deficiency indices define the dimension of the solution spaces of the equations lu = iy and lu = −iy . The goal of the present paper is to study the deficiency indices of a class of differential operators generated on the semiaxis by a self-adjoint differential expression of odd order with complex-valued coefficients. We consider the equation

On the Similarity of \(J \)-Self-Adjoint Differential Operators of Odd Order to Normal Operators

On the Similarity of \(J \)-Self-Adjoint Differential Operators of Odd Order to Normal Operators

Biorthogonal wavelets adapted to integral operators and their applications

First, we construct biorthogonal wavelets adapted to some integral operators, of which the representation matrices by the biorthogonal wavelets are block diagonal. The construction method is applied to the Riesz potential, the first derivative of Hilbert transform as well as the Abel transform. We also show that the Hilbert transform and the differential operators of odd-order, of which the block-diagonal representation matrices can not be constructed by this method, can be decomposed into a sum of operators to which our method is applicable. This decomposition gives a numerical solver for these operators, the number of operations of which is of O(n).

Regularity of self-adjoint boundary conditions

It is shown that self-adjoint boundary conditions for ordinary differential operators of odd order are regular in Birkhoff's sense. A similar result, for differential operators of even order, was proved by a different method by Salaff. In short, Kamke's hypothesis about the regularity of self-adjoint boundary conditions is completely confirmed.

Multiplicity of the spectrum of a self-adjoint ordinary differential operator of odd order

A bound is obtained for the multiplicity of the spectrum of the self-adjoint operator generated by a singular ordinary differential operatorl of odd order in the Hubert space ℒ2 in terms of solutions of the differential equationl[y]=λy.

Groups Whose Commutator Subgroups are of Order Two

If the commutator subgroup of a group G is of order two the commutators of G are invariant and hence every operator of odd order appears in the central of G since such an operator could not be transformed into an operator of even order. It therefore results that when G involves operators of odd order it is the direct product of a group of order 2m and of an abelian group of odd order. It is desirable to exclude direct products in what follows since all except one of the factor groups would be abelian. Hence we shall assume hereafter that the order of G is 2m and that G is not a direct product. The central of G includes the squares of all the operators of G and hence the central quotient group of G is abelian and of type ln, where n is even since the subgroup composed of all of the operators of G which are commutative with two of its iion-commutative operators is of index 4 under G. It is possible to construct as follows a G whose central is an arbitrary abelian group. If tl, t2, * * *, tI is a set of restricted independent generators of this abelian group we divide these operators into distinct pairs when 1 is even or we divide 1 1 of them into distinct pairs when I is odd. In the former case we construct two operators whose squares are the operators of such a pair and that one of these two operators generates the commutator subgroup. Each of these two operators may be assumed to transform the other into itself multiplied by the commutator of order 2. The remaining pairs of independent generators may be assumed to be such that none of them generates the commutator subgroup but that each of the operators of a pair is the square of an operator of G and that two such operators are again non-commutative but are commutative with all of the other operators thus constructed. We thus arrive at a G which has the given abelian group for its central and is not a direct product of two groups since its central is generated by the squares of its operators. When I is odd we may proceed similarly with the exception that the operator which does not appear in a pair may be assumed to be the only one of the set of independent generators which separately

On the group of isomorphisms of a certain extension of an abelian group

In 1908 Professor G. A. Miller showed that if an abelian group H which involves operators whose orders exceed 2 is extended by means of an operator of order 2 which transforms each operator of H into its inverse, then the group of isomorphisms of this extended group is the holomorph of H.t The present paper discusses an elaboration of the idea embodied in Professor Miller's thoerem, the successive developments taking an abelian group H each of more general type so that in toto it is proved that if G is formed by extending H which has operators of order > 2 by a certain operator from its group of isomorphisms which transforms every one of its operators into the same power of itself and which is commutative with no operator of odd order in it, then the group of isomorphisms of G is the holomorph of H, and is a complete group if H is of odd order. If H contains no operators of even order, the certain operator is any operator from the group of isomorphisms of H effecting the stated automorphism; if H contains no operator of odd order, the certain operator must transform every operator of H into its inverse; if H contains operators of both even and odd orders (a) the order of the automorphism of the operators of odd order effected by the extending operator is to be divisible by 2n where H contains an operator of order 2n but none of order 2n+1, or else (b) the extending operator transforms into its inverse every one of H's operators whose order is a power of 2. Obviously it is not necessary that the extending operator be from its group of isomorphisms, but that it have certain properties possessed by this operator; thus, besides effecting the automorphism required, its first power commutative with all the operators of H must be the identity (from which follows that its first power appearing in H is the identity). The general method used in establishing each of the successive theorems is the same; two important steps in each proof are showing that H is character-

Groups containing only three operators which are squares

Every group besides the abelian group of order 2a and of type (1, 1, 1, ***) contains at least two operators which are squares of other operators in the group. If there are only two such operators they constitute an invariant subgroup and the corresponding quotient group contains only one operator which is a square. All the groups which satisfy this condition have recently been determined.t The present paper is devoted to a complete determination of the groups involving just three operators which are squares. As the idenitity is its own square such a group ( G) contains only two operators besides the identity which have the property in question. If the order (g) of G is divisible by an odd number, this number is 3 and G includes only one subgroup of this order, since every operator of odd order is the square of some power of itself. As suich a group cannot contain an operator of order 4 and its order is 3.2a, it niust be the direct product of the abelian grouip of order 2a and of type (1 , 1, 1, * * *), and the cycle group of order 3; or one of its subgroups of order 3.2a-1 must be of this type. In the latter case, G is evidently the direct product of the symmetric group of order 6 and the abelian group of order 2a-1 and of type ( 1, 1, 1, *) . In what follows, these trivial cases will not be considered. Hence g will always be of the form 2a and G contains only operators of orders two and four in addition to the identity. Moreover, each of the two operators (t,, t2) which are squares but are not the identity is of order two and hence it is the square of some operators of order four. We shall first prove that t1, t2 are invariant under G.