(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2015)
We were invited by the Organizing Committee of the Mathematical Conference dedicated to Professor Ivan Prodanov on the occasion of the 60-th anniversary of his birth and the 10-th anniversary of his death, held on May 16, 1995 at the Faculty of Mathematics and Informatics of the Sofia University “St. Kliment Ohridski”, to give a talk on his investigations in the theory of abstract spectra. All of his results in this area were announced in a short paper published in the journal “Trudy Mat. Inst. Steklova”, 154, 1983, 200–208, and, as far as we know, their proofs were never written by him in the form of a manuscript, preprint or paper. The very incomplete notes which we have from Prodanov’s talks on the Seminar on Spectra, organized by him in the academic year 1979/80 at the Faculty of Mathematics of the University of Sofia, seem to be the only trace of a small part of the proofs of some of the results from the cited above paper. Since the untimely death of Ivan Prodanov withheld him from preparing the full version of this paper and since, in our opinion, it contains interesting and important results, we undertook the task of writing a full version of it and thus making the results from it known to the mathematical community. So, the aim of this paper is to supply with proofs the results of Ivan Prodanov announced in the cited above paper, but we added also a small amount of new results.The full responsibility for the correctness of the proofs of the assertions presented below in this work is taken by us; just for this reason our names appear as authors of the present paper. The talks of the participants of the conference had to be published at a special volume of the Annuaire de l’Universite de Sofia “St. Kliment Ohridski, but this never happened. That is why we have decided to publish our work separately. Since our files were lost and we had to write them once more, the paper appears only now.