A general HELP inequality connected with symmetric operators

In this paper, a general HELP (Hardy-Everitt-Littlewood-Pólya) inequality is considered which is connected with a symmetric operator in a Hilbert space and abstract boundary mappings. A criterion for the validity of such an inequality in terms of the abstract Titchmarsh-Weyl function is proved and applied to Sturm-Liouville operators, difference operators, a Hamiltonian system and a block operator matrix

Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E hooked right arrow F are symmetric sequence spaces with E ≠ l(1) and F ≠ l c(0) and l(∞) then there exist a intermediate symmetric sequence space G such that E hooked right arrow G hooked right arrow F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c(o) and l(1) inside the class of all symmetric sequence spaces are given

Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function

summary:This work is concerned with the eigenvalue problem for a monotone and homogenous self-mapping $f$ of a finite dimensional positive cone. Paralleling the classical analysis of the (linear) Perron–Frobenius theorem, a verifiable communication condition is formulated in terms of the successive compositions of $f$, and under such a condition it is shown that the upper eigenspaces of $f$ are bounded in the projective sense, a property that yields the existence of a nonlinear eigenvalue as well as the projective boundedness of the corresponding eigenspace. The relation of the communication property studied in this note with the idea of indecomposability is briefly discussed