Overview of published works

The bibliographical references in the text below are found in the enclosed list of works.

The theory of principal modules and its applications to the study of regular operators: [16] - [19], [21], [23], [24], [28], [32], [34], [35], [52], the book [3].

The notion of a principal module was introduced in the INCREST preprint 62/1980 Strongly lattice ordered modules over function algebras (published at a later date as [17] and [18]). The theory of principal modules comprises two components : the study of algebraic and topological properties of these modules; the study of principal modules of linear operators.

In the INCREST preprint 100/1981 Extension theorems for strongly lattice - ordered modules and applications to linear operators the principality permanence theorem was proved. The theorem was also stated in [16], [19], [21]. A systematic study of the applications of the permanence theorem to majorization problems in operator theory has been done. These problems, which received consideration from the part of many researchers during the past decade, had as a general theme the discovery of subclasses C of L_r(E,F) with the property that the relations 0 <= S <= T, S belongs to L_r(E,F) and T belongs to C imply S belongs to C. As classical examples of majorization theorems, one could mention the theorem of Dodds and Fremlin on compact operators and the theorem of Lozanovskii, Buhvalov and Schep on band properties of the space of integral operators. Although the statements of these two theorems have in common the aspect proper to a majorization problem, their original proofs have very few things in common. By using the permanence theorem of the theory of principal modules it was possible to elaborate a unified method for the study of majorization problems which allowed to redemonstrate, on a common basis, most of the known majorization theorems (in particular the ones mentioned above) and to find new results.

Among the new results found with the aid of that method, one mentions the results in [19] and [28] on approximable operators and the results in [19], [21] and [28] on the relation between the order ideal and the closed algebraic ideal generated by a regular operator.

The theory of oru-compact operators: [29], [33], [36].

A class of regular operators which possess moduli even if the target space is not order complete.

General theory, the relation with almost integral operators in the sense of G. Ya. Lozanovskii, the relation with integral operators.

The theory of strongly latticial classes: [35].

Methods of construction of spaces of regular operators which possess moduli and their applications to majorization problems.

Contributions to the study of integral operators: [29], [34], [36], [40], [41], [52].

Contributions to the problems of the intrinsic characterization of integral operators and of the characterization of some particular classes of integral operators (Carleman, Hilbert-Schmidt) in terms of algebraic ideal properties.

The theory of Convolution Type Operators, which provides a unified framework for the study of order properties of several classes of operators including integral, convolution, change of variable operators and Radon transforms.

Contributions to the theory of factorization of compact operators: [39], [44].

Contributions to the study of abstract Riesz spaces and of the operators on these spaces.

The Riesz decomposition property for complex Riesz spaces [7], the construction of the perfect M-tensor product [16], the relation between almost integral operators and finite rank operators [34], the extension of Riesz space valued operators [6], [10], the extension of operators invariant with respect to a semigroup [5], [10], the relation between the weak Fatou property and the Egoroff property [22], the integral representation of regular operators (the book [3]).

Contributions to the mathematical theory of music: [11] - [15], [20], [25], [27], [30], [31], [38], [43], [46] - [49], [54], the textbook [2].

The applications of algebraic structures (especially groups and their actions) and of discrete harmonic analysis to music theory, searching for adequate mathematical modeling of some classes of musical phenomena, especially those involving periodicity. In this context a model [26] intended for the study of periodic rhythm was proposed.

The study in four parts Supplementary Sets and Regular Complementary Unending Canons, published in Perspectives of New Music, is concerned with the applications of discrete harmonic analysis (convolution and Fourier transform on the groups Z_n ) to the theory of unending rhythmic canons. The notion of supplementary sets turned out to be a mathematical model adequate for the study of the regularity properties and the construction of the named canons.

Other contributions.

Measure theory, differential manifolds, topological groups, the theory of Alfsen and Effros, operatorial characterizations of Hilbert spaces, reflexive spaces of operators on Hilbert spaces.