# Shizuo Kakutani

### Quick Info

Osaka, Japan

New Haven, Connecticut, USA

**Shizuo Kakutani**was a Japanese-born American mathematician who worked in probability theory, ergodic theory and complex analysis, and is best known for his fixed-point theorem.

### Biography

**Shizuo Kakutani**'s father, Kakujiro Kakutani, was a lawyer. Shizuo was the youngest of the two sons in the family, the eldest being Seiichi who was eight years older. Seiichi studied physics at Kyoto University and it was through him that Shizuo was first introduced to mathematics. When he was about nine years old his elder brother, on one of his frequent visits back to his home while he studied at university, would explain mathematical ideas to him.

Shizuo was fascinated and was enthusiastic to learn more mathematics. However two factors conspired to make this impossible. The first problem was that Kakujiro Kakutani had made the decision that one of his two sons would follow him into law and take over his practice in due course. Clearly Seiichi was training to be a physicist and not studying law so Shizuo would have to be the one to follow his father. The mathematics lessons from Seiichi were tragically cut short when he died of typhoid fever at the age of twenty.

After completing his middle school, Shizuo entered Konan High School in Kobe to prepare for his university studies. At this stage he had to choose between arts subjects or sciences and his father gave him no choice since studying law at university required him to graduate from high school which qualifications in literature and arts. By the time Shizuo graduated, his father relented seeing that his son was so keen to study mathematics at university. However, Shizuo was now not qualified to enter a mathematics course at either the University of Tokyo or Kyoto University since these had absolute rules regarding entry qualifications.

There was one possible route for Shizuo which was to enter Tohoku University in Sendai. This did not prevent those without a science qualification entering a mathematics course but it gave preference to those the scientific training. Kakutani applied but there were only fifteen places, and seventeen applicants. Of the seventeen, exactly fifteen had the science qualification from high school so it looked like an easy task to decide to admit those and to turn down Kakutani. However, after due consideration it was decided to admit all seventeen applicants and Kakutani had scrape through.

At Tohoku University Kakutani was introduced to the theory of analytic functions. He read various classic texts including those of Stone and Banach and by the time of his graduation at the end of the three year course he had a good foundation in modern analysis. He was appointed as a teaching assistant at Osaka University in 1934 where he collaborated with K Yosida on a paper on Nevanlinna theory. Hajian and Ito write in [3]:-

During his years at Osaka University Shizuo Kakutani had already established himself as a research mathematician by publishing a number of papers in functional analysis and ergodic theory, and the 1937 paper in the Japanese Journal on Mathematics on Riemann surfaces. It was this work which was later to become the main part of his doctoral dissertation. it was also this paper that caught the attention of Weyl ...Indeed on the strength of this work Weyl invited Kakutani in 1940 to spend two years at the Institute for Advanced Study at Princeton. Kakutani not only took great interest in the work of Weyl's group at Princeton but also the group of mathematicians working with von Neumann on measure theory and ergodic theory. He met many other young mathematicians at Princeton who would influence him such as Ambrose, Halmos, Doob, and ErdÅ‘s. Kakutani also made many visits during which he met mathematicians such as Garrett Birkhoff, G D Birkhoff, Stone, Wiener, and Hille.

In December 1941 with Kakutani still studying at Princeton, war broke out between the United States and Japan with the entry of the U.S.A. into the Second World War. Of course this put Kakutani in a difficult position for he was now a guest in a country at war with his own. S Eigen says in [2]:

With the outbreak of war he was given the option of staying at the Institute or returning to Japan. He chose to return because he was concerned about his mother. So he was put on a Swedish ship which sailed across the Atlantic, down around the Cape, and up to Madagascar, or thereabouts, where he and other Japanese were traded for Americans aboard a ship from Japan.On his return to Japan Kakutani accepted the appointment as assistant professor at Osaka University. He also continued his collaboration with Yosida, who by this time was at Nagoya University, and began a new collaboration with Yosida's colleague Kiyosi Ito at Nagoya. The later war years were particularly difficult ones in Japan and many Japanese mathematicians failed to keep their research going through the difficulties of these times. Kakutani, however, managed to continue to produce a stream of papers containing highly original ideas.

The trip across the Atlantic was long and hard. There was the constant fear of being torpedoed by the Germans. What, you may wonder, did Kakutani do. He proved theorems. Every day, he sat on deck and worked on his mathematics. Every night, he took his latest theorem, put it in a bottle and threw it overboard. Each one contained the instruction that if found it should be sent to the Institute in Princeton. To this day, not a single letter has been received.

In 1948 Kakutani was again invited to the Institute for Advanced Study at Princeton. In the summer of that 1949 he worked at the University of Illinois, then later that year accepted the offer of an appointment at Yale University. On one of his frequent visits to New York, Kakutani met Kay Uchida and they were married in 1952; they had one daughter Michiko.

Kakutani contributed to several areas of mathematics. Among the areas on which he has written papers we must mention: complex analysis, topological groups, fixed point theorems, Banach spaces and Hilbert spaces, Markov processes, measure theory, flows, Brownian motion, and ergodic theory.

Kakutani was to remain at Yale until he retired in 1982. In June of that year he received the Academy Award and the Imperial Award of the Academy of Japan.

As to Kakutani's personality, Hajian and Ito write in [3]:-

Professor Kakutani is a gentleman and a scholar of the old school. His mild manner, gentle graciousness, and total dedication to mathematics leave an indelible impression on all who have gotten to know him.

### References (show)

- R R Kallman (ed.),
*Shizuo Kakutani : Selected papers*(2 Vols) (Boston, MA, 1986). - A Hajian and Y Ito, Shizuo Kakutani, in
*Shizuo Kakutani : Selected papers***1**(Boston, MA, 1986). - K Yosida and Y Ito, Award of Academy-Imperial Prize to Shizuo Kakutani (Japanese),
*Sugaku***34**(4) (1982), 351-354.

### Additional Resources (show)

Other websites about Shizuo Kakutani:

### Honours (show)

Honours awarded to Shizuo Kakutani

Written by J J O'Connor and E F Robertson

Last Update September 2001

Last Update September 2001