**Ivan Vidav**was born in a village named Opcine (Italian Villa Opicina), near Trieste, which at the time of his birth was in the Austro-Hungarian Empire but, soon after the end of World War I, it came under the rule of the Kingdom of Italy. Ivan was the youngest of four sons of a small-time farmer, also called Ivan Vidav, and his wife Josipina Sosic. Josipina had first married Ivan's uncle before her marriage to Ivan. Ivan junior, the subject of this biography, also had a step-sister who was fourteen years his elder.

In 1920 the Vidav family moved to the vicinity of Maribor in Yugoslavia. Because of poor management of their small farm in the difficult conditions created by the worldwide economic crisis at the end of the 1920s, the family found itself in a very bad material situation just when Ivan went to secondary school. He was studying at the grammar school in Maribor from 1929 until 1937. In the higher grades he tutored other pupils in mathematics in order to earn a living and even earned enough to help support his family. His family suffered heavy losses during World War II; only his mother, despite being confined in a Nazi concentration camp, Ivan, and his step-sister survived.

In 1937 Vidav began to study mathematics at the University of Ljubljana. The leading professor there was Josip Plemelj who was renowned for solving the Riemann-Hilbert problem (i.e. the 21st problem on Hilbert's famous list). The professor soon realized the outstanding mathematical ability of his student. They often discussed mathematics of all kinds and on one occasion, after his lecture in 1940, Plemelj mentioned to Vidav an open problem concerning homogeneous linear differential equations. Then, during the summer holidays of 1940, the 22 year old Vidav solved the problem and passed the solution to professor Plemelj at their first meeting at the beginning of the autumn semester. We can see Plemelj's big surprise by quoting his own words:-

This first original scientific work of Vidav, in which he considered the so-called Fuchsian differential equation with five singular points, became the basis of his dissertation entitledI hadn't ever thought that he would even take the problem into consideration and by no means to be able to succeed. ... His work identifies and explains fully the various problems in the theory of linear differential equations, problems that intrigued many mathematicians but none was able to penetrate to their core. ... Every mathematician would be proud to have such a masterpiece among his bibliography. Although this is the first paper of Ivan Vidav, it contains no beginner's deficiency; on the contrary, it has the flavour of a mature mathematician's work.

*The theorems of Klein in the theory of linear differential equations*. It is worth mentioning that Vidav was awarded his diploma and the Ph.D. degree (with Plemelj as advisor) in the same year, 1941, just when the war was spreading to South-East Europe.

He studied in Rome in 1941-42. After the war, in 1946, he became an Assistant Professor and, in 1853, a Full Professor at the University of Ljubljana. At first he continued working on problems in differential equations. For instance, he participated at the 11^{th} International Congress of Mathematicians at Harvard University in Cambridge, USA, from 30 August to 6 September in 1950, where he gave a short report on his own results on Fuchsian equations with five or six singularities. During his rapid academic ascent in the next few years he made several short visits to Paris, where he met Szolem Mandelbrojt, who influenced his later research in approximation theory. In this period he published in *Acta Mathematica* and *Comptes rendues de l'Academie des sciences *(in French) as well as in *Mathematische Zeitschrift* (in German) and in some local Yugoslavian journals. For a selection of some of his papers, see THIS LINK.

In the mid-fifties, he became interested in functional analysis. In this area he contributed to some of the basic notions and results in the theory of Banach algebras and C*-algebras. He wrote a fundamental paper on metric characterization of Hermitian elements in a Banach algebra, published as *Eine metrische Kennzeichnung der selbstadjungierten Operatoren* Ⓣ in *Mathematische Zeitschrift* in 1956. The main result is a characterization of C*-algebras among Banach algebras, obtained under some additional assumptions which were removed a decade later by E Berkson, B W Glickfeld and T W Palmer (independently). The final result is called the 'Vidav-Palmer theorem'. Some other notions and results are named after Vidav: a definition of Hermitian elements, a lemma about their numerical range, an involution, and an algebra. In the following years Vidav published further papers on functional analysis, some in German, some in French, and later some in English. Because of his international success in this area he was elected as a member of Slovenian Academy of Arts and Sciences in 1962.

In the mid-sixties, physicists attracted Vidav to consider various problems in neutron transport theory. He published several results on strongly continuous semigroups of operators, connected with the linear Boltzmann equation (mostly in *Journal of Mathematical Analysis and Applications*), sometimes in cooperation with other mathematicians and physicists. For his work in this applied area of functional analysis he received the most prestigious Slovene scientific award at the time in 1970. He continued to undertake research also on the isometric structure of Banach spaces and on analytic operator functions. During his almost three decades of active mathematical investigations in functional analysis, he became a thesis advisor to sixteen Ph.D. students. He encouraged them to explore several different areas of mathematics.

Vidav became, as a long-time chair, the master architect of the Mathematics Department at the University of Ljubljana after 1960. He was responsible for all major curriculum reforms in the education of young mathematicians in the sixties and the seventies, modernizing the study of mathematics in Ljubljana to become comparable to foreign universities. He initiated many new trends and improvements in the teaching process, including the opening of regular graduate studies in mathematics at the beginning of the seventies. He was often the first and the main lecturer, an excellent one; he never used notes in the classroom. He wrote several textbooks for students of mathematics, e.g. a classic three-volume calculus course (*Higher mathematics*) and standard monographs on algebra, affine and projective geometry, differential geometry, elliptical curves and functions, and several graduate texts on Banach algebras, *C**-algebras, linear operators in Banach spaces, group theory, and *K*-theory (all in Slovenian). He also wrote several little gems of books popularizing mathematics, e.g. *Solved and unsolved problems in mathematics* (in Slovenian).

After his retirement in 1986 he continued for twenty years still publishing articles in these areas, mostly in mathematical journals in Slovenia, such as *Obzornik za matematiko in fiziko * Ⓣ and *Presek* Ⓣ. Some of these articles, primarily intended to introduce mathematics to youngsters, contain original mathematical works.

Besides his scientific and pedagogical obligations he was an active member of the Society of Mathematicians, Physicists and Astronomers of Slovenia from the foundation of the Society, serving as its chairman from 1951 to 1954, its vice chairman from 1955 to 1959, for many years the organizer of the national competition in mathematics for grammar school pupils, the main editor of several book series published by the Society, and more. He became an honorary member of this Society in 1988. He was also awarded some of the highest academic honours by the University of Ljubljana and also some of the highest state awards in the area of scientific research and developmental activities in Slovenia as well as in former Yugoslavia.

Professor Vidav was a leading scientific and moral authority in Slovenian mathematics after World War Two until his retirement. He was a teacher of many generations of students and a master of mathematical writing. Throughout his active life he felt a responsibility for the development of mathematics at the University of Ljubljana and for the whole mathematical community in Slovenia. Moreover, with his high standards and an almost proverbial modesty, he remained a man of strong personal integrity.

**Article by:** *Milan Hladnik* [Thanks to Tomo Pisanski for sending us this biography]

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