During the last few years several articles on asymptotic martingales (amarts) have appeared. The first unified treatment was given by Edgar and Sucheston in [7], where further references can be found. The purpose of this paper is to add some further results to the theory of amarts.

The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2, 6, 7].

All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left (right) stable if

(ρ): for every ring R and every left (right) ideal I of R it follows S(I)⊆S(R).

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following property

Conversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

This paper results from the design and development of computer software for lengthy computations with Cayley's algebra C over the field of reals. The algebra C is 8-dimensional over the reals and is not associative. Integer elements of C are defined and can be stored as integer arrays. The problem of solving linear equations αξ = β in C is implemented by using the equation where is the conjugate and N(α) is the non-zero norm of α. Programming multiplication of Cayley numbers is comparable in difficulty to programming matrix multiplication for matrices with eight rows and eight columns.