Kakutani's research career extended over half a century and more than a hundred scientific papers. His work was noted for its versatility, originality and clarity of exposition. It covered a broad range of fields, most notably probability theory, fixed-point theorems, ergodic theory and complex analysis.

Kakutani was the first to note the fundamental link between probability theory and potential theory, in 1944. It is remarkable that electromagnetism and gravity -- the two best known of the four fundamental forces in nature -- are governed by very similar laws. The resulting subject, potential theory, dates from the early 19th century.

Brownian motion -- the random motion of fluctuating particles (of dust in a sunbeam, say) -- dates back to the same period, though its mathematical basis came only in the early 20th century. After Kakutani's fundamental work on Brownian motion and potential theory, the subject of probabilistic potential theory has had a great impact, most notably in the hands of his near-contemporary J. L. Doob (obituary, July 26, 2004).

Kakutani's work on fixed-point theorems also had great impact. In many situations involving motion, one can prove from quite weak assumptions that there must be points fixed under the motion (as in the "eye of a storm", for example). Kakutani's fixed-point theorem of 1941 has been widely used, not only within mathematics, but also within economics, where fixed points correspond to equilibriums reflecting balance of supply and demand.

Two applications of Kakutani's work here became famous in economics and led to nobel prizes -- to John nash for his concept of nash equilibriums in game theory, and to Arrow and Debreu for their work on economic equilibriums in competitive economies. Kakutani could not be similarly honoured himself, because there is no nobel Prize for Mathematics.

Kakutani also did important work in several other areas, mainly in mathematical analysis. He worked on ergodic theory (which studies limiting properties, as in physical systems approaching equilibrium), functional analysis (the mathematics of infinite-dimensional systems, such as arise in quantum mechanics and elsewhere), complex analysis (as with his early work on Riemann surfaces) and other fields.

Shizuo Kakutani was born in Osaka, Japan, in 1911. His father was a distinguished lawyer, who wished one of his two sons to follow him into his legal practice. The elder son Seiichi studied physics, but passed on to Shizuo his enthusiasm for mathematics and science.

After Seiichi's early death from typhoid fever, his father wisely allowed Shizuo to follow his obvious bent for mathematics. Equally wisely, Tohoku University in Sendai admitted him, despite his not having a science diploma from high school.

Kakutani flourished in the lively mathematical environment he found there. He became a teaching assistant at the newly formed mathematics department at Osaka University in 1934. His first paper appeared in 1935; his seventh paper (on Riemann surfaces in complex analysis) both formed the basis of his doctoral thesis and led to his being invited (by Hermann Weyl) to the prestigious Institute for Advanced Study in Princeton, new Jersey, which he visited from 1940 until 1942.

Despite the outbreak of war between the USA and Japan after the attack on Pearl Harbor on December 7, 1941, Kakutani was allowed to complete his visit, although thousands of US citizens of Japanese descent were interned on the West coast as a security risk. He returned to Japan as an established mathematician of world class, and was appointed Assistant Professor at Osaka University.

Kakutani returned to the USA, this time permanently, in 1948. He visited Princeton for another year, and then joined the mathematics department at Yale University, where he spent 33 productive years until his retirement. He married Kay Uchida of new York City in 1952; they had a daughter, Michiko.

Kakutani was widely honoured. In 1982 he received the Academy Award and the Imperial Award of the Academy of Japan for his mathematical achievements. He had many doctoral students, a number of whom have become major mathematicians, and collaborated widely.

Mathematicians have the reputation of being solitary, difficult people. Kakutani was quite the opposite. He was mathematically gregarious, and loved to collaborate, though he did not care to travel -- so his collaborators tended to visit him. He was a very pleasant and courteous man, a dedicated mathematician, a scholar and a gentleman. He will be warmly remembered for his human qualities by those who knew him, and by the scientific world for the ongoing importance of his life's work.

Shizuo Kakutani, mathematician, was born on August 28, 1911. He died on August 17, 2004, aged 92.

© The Times, 2004