In 1999-2000 we considered maximal operator semigroups, bounded in a certain sense, on real or complex vector spaces. We proved that such as semigroup is equal to the semigroup of all contractions with respect to some quasinorm. Using a recent remarkable theorem of Turovskii we extended the famous result of de Pagter to semigroups by showing that every multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two has a common non-trivial invariant closed ideal. We also constructed an irreducible multiplicative semigroup of square 0 positive operators acting on L 2(0,1). Working on multiparameter spectral problem in the years 2001-03 we focused on the associated system of commuting operators. In the finite-dimensional setup we proved that the algebra generated by the commuting operators is always an Artin-Gorenstein algebra. These algebras have been extensively studied by the specialists in the field of commutative algebra. We expect that their results will influence the development of the multiparameter spectral theory. We also proved a generalization of the classical Cayley-Hamilton Theorem for multiparameter systems. In 2001 we introduced the notion of representation of a module. We defined derivations on modules and, for Banach modules, we proved two theorems that are module variants of Gleason-Kahana-Zelazko Theorem and Singer-Wermer Theorem, respectively. We extended well known results about decomposability of multipliers on semisimple commutative Banach algebras to the realm of Banach modules over spectrally separable algebras. We introduced the class of strongly harmonic operators and gave a partial answer to the question, posed by Laursen and Neumann in their monograph, on the relation between the local spectra of an elementary operator and the local spectra of its coefficients. We studied operators on left Banach modules and introduced the class of simple multipliers. We showed that these operators behave, from the point of view of the local spectral theory, similar as multipliers on semisimple Banach algebras. In 2001 we proved the Intersection Theorem for orderings of higher level on Ore domains. We applied the results to noncommutative real algebraic geometry (to prove higher level Positivstellensatz for noncommutative noetherian rings) and to the theory of quadratic forms (to study sums of permuted products of n-th powers in noncommutative noetherian rings.) During 2001-2003 we were able to completely characterise additive mappings, defined on the algebra of finite-rank operators on a fixed Banach space, and which do not increase rank-one operators. The linear version of our result is well-known, and proved to be an important tool in investigating linear preservers. In the years 2000-2003 we studied orthogonality of the range and the kernel of elementary operators with respect to operator norm and von Neumann-Schatten norms. We focused primarily on elementary operators with normal coefficients and proved some new characterisations of the kernel of elementary operator. Using Frechet differentiability of von Neumann-Schatten norms we were able to characterise operators orthogonal to the range of arbitrary elementary operators.