Vol 241, No 4 (2019)

The eigenfunction problem for a scalar Euler operator leads to an ordinary differential equation, which is an analog of higher-order Bessel equations. Its solutions are expressed through elementary functions in the case where the corresponding Euler operator can be factorized in a certain appropriate way. We obtain a formula describing such solutions. We consider the problem on common eigenfunctions of two Euler operators and present commuting Euler operators of orders 4, 6, and 10 and a formula for their common eigenfunction and also commuting operators of orders 6 and 9.

In this paper, we study a system of nonlinear equations on a square graph related to the affine algebra A⁽¹⁾₁ . This system is the simplest representative of the class of discrete systems corresponding to affine Lie algebras. We find the Lax representation and construct hierarchies of higher symmetries. In neighborhoods of singular points ⋋ = 0 and ⋋ = ∞, we construct formal asymptotic expansions of eigenfunctions of the Lax pair and, based on these expansions, find series of local conservation laws for the system considered.

We discuss computational aspects of the spectral problem for the curl of a vector field that allow one to find tangent fields to coordinate surfaces of a given curvilinear coordinate system.

Let \( \mathcal{M} \) be the von Neumann algebra of operators in a Hilbert space \( \mathcal{H} \) and τ be an exact normal semi-finite trace on \( \mathcal{M} \). We obtain inequalities for permutations of products of τ-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of τ -measurable operators and a sufficient condition of orthogonality of certain nonnegative τ-measurable operators. We state sufficient conditions of the τ –compactness of products of self-adjoint τ -measurable operators and obtain a criterion of the τ -compactness of the product of a nonnegative τ-measurable operator and an arbitrary τ -measurable operator. We present an example that shows that the nonnegativity of one of the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from \( \mathcal{M} \) . All results are new for the *-algebra \( \mathcal{B} \)(\( \mathcal{H} \)) of all bounded linear operators in \( \mathcal{H} \) endowed with the canonical trace τ = tr.

We show that ordinal isomorphisms of orthogonal measures on state spaces of operator algebras equipped with the Choquet order are generated by Jordan isomorphisms of associated von Neumann algebras. This yields a new Jordan invariant of σ-finite von Neumann algebras in terms of decompositions of states.