For given square matrices A and B we denote by Y = AB -BA and by Z = AY -YA. It is well known that if A and Y commute, i.e., if Z = 0, then Y is a nilpotent matrix. In this note we show that the same is true if YZ = ZY . We also generalize this result by using commutators of higher order.

Denote by $C$ the commutator $AB-BA$ of two bounded operators $A$ and $B$ acting on a locally convex topological vector space. If $AC-CA=0$, we show that $C$ is a quasinilpotent operator and we prove that if $AC-CA$ is a compact operator, then $C$ is a Riesz operator.

We investigate some properties of an algebraic operator A on a general vector space X and especially in the case when X is a locally convex space. We prove that A is always hyporeflexive and that it is reflexive if its minimal polynomial is simple. Moreover, we show that this condition is necessary and sufficient for the reflexivity of the commutant of A. We also show that the second commutant of A is equal to the algebra generated by A and the identity operator. In the last section we prove that every locally algebraic operator acting on a Fréchet space is algebraic, and that an operator which is a finite rank perturbation of an algebraic operator is again algebraic.

Some simple algorithms are given for evaluation of determinants using only the second-order subdeterminants together with some illustrative examples. The traditional methods for hand-calculation of the determinant of an n × n ma- trix are based either on simplifying the matrix by performing elementary row or column operations or on expansion by minors along some row or column. A brief overview of the theory of determinants can be found, for example, in (6) and (7). There are many commonly-used computer packages such as Mathematica or Mat- lab in which the algorithms to find the determinant of a matrix are based on factorisation in a product of lower and upper matrices. The purpose of this note is to present some methods for evaluation of determinants using only second-order subdeterminants.

Some results concerning reducibility and triangularizability of some sets of algebraic and of compact operators on locally convex spaces are given.

A generalization of some results from normed spaces, concerning reducibility and triangularizability of semigroups and algebras of operators, to locally convex spaces is given.

Some results concerning hyperinvariant subspaces of some operators on locally convex spaces are considered. Denseness of a class of operators which have a hyperinvariant subspace in the algebra of locally bounded operators is proved.

The invariant subspace problem for some operators and some operator alge- bras acting on a locally convex space is studied.

The spatial numerical range for a class of operators on locally convex space was studied by Giles, Joseph, Koehler and Sims in (3). The purpose of this paper is to consider some additional properties of the numerical range on locally convex and especially on H-locally convex spaces.

Some simple algorithms are given for evaluation of determinants using only the second-order subdeterminants together with some illustrative examples. The traditional methods for hand-calculation of the determinant of an n × n matrix are based either on simplifying the matrix by performing elementary row or column operations or on expansion by minors along some row or column. A brief overview of the theory of determinants can be found, for example, in [6] and [7]. There are many commonly-used computer packages such as Mathematica or Matlab in which the algorithms to find the determinant of a matrix are based on factorisation in a product of lower and upper matrices. The purpose of this note is to present some methods for evaluation of determinants using only second-order subdeterminants.