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A self-adjoint second-order differential expression with an unbounded operator coefficient is considered in the space of vector functions. The domain of definition of the minimum and maximum operators generated by this expression is investigated, it is shown that any generalized resolvent of the minimum operator is an integral operator, and expansion in eigenfunctions is performed.

A theorem is proved on the inversion of operators of the form (d/dt) + A, where A is a linear operator in a space of vector functions bounded on the axes.

We introduce a class of linear operators, containing the positive operators, in which P. P. Korovkin's theorem on the conditions and order of convergence of positive operators hold. We consider also the class of linear operators, convergent to the derivative, in which similar theorems hold.

Theorems of M. A. Krasnosel'skii are generalized which concern fixed points of completely continuous operators which compress or expand a cone, and a topological principle for fixed points in the case of P-compact operators.

The definition given by P. P. Korovkin of operators of the class Sm and conditions for the convergence of these operators to the identity operator are extended to apply to regular operators from a K-space R0 with a unit, into a K-space R1 where R0 and R1 are normally contained in the union of the spaces S [a, b] and s.

In generalizing a series of known results, the following theorem is proved: If K is a continuous linear operator mapping E0 into F0 and E1 into F1 (where E0, E1 and F0, F1, being ideal spaces, are Banach lattices of functions defined on Ω1 and Ω2 respectively), then for any λ. e (0, 1) K maps E 0 1−λ E 1 λ into [(F 0 ′ )1−λ(F 1 ′ )λ]′ and is continuous; for suitably chosen norms in the spacesE 0 1−λ E 1 λ and [(F 0 ′ )1−λ(F 1 ′ )λ] the norm of K is a logarithmically convex function of λ. Six titles are cited in the bibliography.

This paper describes a class of linear operators A which have the property that ∥I+A∥ = 1 + ∥ A ∥, and which act in the space C of abstract continuous functions. In particular, to such operators, called rammers, are referred completely continuous operators and operators acting in C not only from C, but from some essentially broader space. As an application, it is shown that each linear operator B in C, whose norm is 1 and which is the identity of a subspace of finite deficiency, coincides with the identity on the whole space.