# Kosaku Yosida

### Born: 7 February 1909 in Hiroshima, Japan

Died: 20 June 1990 in Tokyo, Japan

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**Kosaku Yosida**was born in Hiroshima, a large industrial city on Honshu Island around 800 km south east of Tokyo. He was brought up in Tokyo where he attended school, then entered the Faculty of Science of Tokyo Imperial University. This university, established in 1877, had been devastated by the Great Kanto earthquake which struck on Saturday, 1 September 1923. Yosida graduated with his first degree in 1931, having taken mathematics as his main subject. He had been taught by Takuji Yosie (1874-1947) and was friendly with fellow student Hidegoro Nakano (1909-1974) who also studied at Tokyo Imperial University and obtained the degree of bachelor of science in March 1933.

Yosida continued to study at Tokyo Imperial University beginning research and publishing papers on meromorphic functions and ordinary differential equations such as:

*On the asymptotic property of the differential equation*

*y*" +

*H*(

*x*)

*y*=

*f*(

*x*,

*y*,

*y*') (1932);

*On the distribution of a-points of solutions for linear differential equation of the second order*(1932);

*On the characteristic function of a transcendental meromorphic solution of an algebraic differential equation of the first order and of the first degree*(1933);

*A remark to a theorem due to Halphen*(1933),

*A note on Riccati's equation*(1933);

*A generalisation of a Malmquist's theorem*(1933); and

*On a class of meromorphic functions*(1934).

In 1933 Yosida was appointed as an Assistant in the Department of Mathematics at Osaka Imperial University. Osaka is on Honshu Island, roughly half way between Hiroshima and Tokyo. The Osaka Imperial University is based on educational institutions dating back to the 18th century but only became a university in 1931, two years before Yosida was appointed there. After one year, he was promoted to Associate Professor.

Moving to Osaka Imperial University led to Yosida changing the direction of his research. Two mathematicians who joined the Department of Mathematics shortly after him and were to strongly influence him were Mitio Nagumo (1905-1995) and Shizuo Kakutani. Nagumo had graduated from Tokyo Imperial University in March 1928 and spent two years at the University of Göttingen, Germany, before being appointed to Osaka Imperial University in March 1934. Kakutani had studied at Tohoku University in Sendai before being appointed as a teaching assistant at Osaka Imperial University in 1934. Yosida became interested in functional analysis through discussions with these two mathematicians. He published several joint papers with Kakutani, for example:

*On meromorphic functions*(1935);

*Application of mean ergodic theorem to the problems of Markoff's process*(1938);

*Markoff process with an enumerable infinite number of possible states*(1939); and

*Birkhoff's ergodic theorem and the maximal ergodic theorem*(1939). In this last mentioned paper the authors prove two theorems, the first being the Ergodic Theorem of G D Birkhoff (in its form given by Andrei Kolmogorov in a 1937 paper), while the second theorem is new and the authors call it the Maximal Ergodic Theorem. This new theorem is stronger than one proved by Norbert Wiener in his 1939 paper

*The Ergodic Theorem*. The authors write:-

Kiyosi Ito writes in [1]:-In the present note, we shall give a direct proof of Theorem2. Our method of proof is a modification of that of Khinchin-Kolmogorov which was used to prove Theorem1; and it is to be noted that we can prove Theorem1(G D Birkhoff's Ergodic Theorem)and Theorem2(Maximal Ergodic Theorem)simultaneously by the same principle without appealing to Maximal Theorem[of Hardy and Littlewood]nor to the Mean Ergodic Theorem[of von Neumann].

In 1939 Yosida was awarded a doctorate from the Osaka University and, in the following year, he spent the autumn quarter of 1940-41 at the University of Washington in the United States:-The most original work during his stay in Osaka(1933-42)is probably a series of papers on the operator-theoretical treatment of ergodic theory, some written jointly with Kakutani.

In 1942, he moved to Nagoya when appointed as a Professor in the Department of Mathematics at Nagoya Imperial University. Nagoya lies between Osaka and Tokyo and became the site of Nagoya Imperial University in 1939, the university being established on a medical school which had existed there since 1871. Yosida's main task at Professor of Mathematics was to promote and expand the newly established Department of Mathematics and this he did very successfully, through much hard work, along with his colleagues. Kiyosi Ito writes in [1]:-The University of Washington announced that Professor Kôsaku Yosida of the Mathematical Institute of Nagoya University has been appointed Walker-Ames Professor for the autumn quarter.

Yosida publishedThe work that made Yosida's name world-famous must be the theory of semigroups and its applications. This theory was published in1948when he was at Nagoya University. It is now called the Hille-Yosida theory .... The influence of this theory was far-reaching: Yosida and many others applied it to numerous areas in analysis such as diffusion equations, Markov processes, hyperbolic equations, and potential theory.

*Topological Analysis*in Japanese in 1951. This book was a major contribution to functional analysis, the subject being known as 'topological analysis' in Japan at this time. Hirotada Anzai writes in a review:-

This book was expanded and modernised by Yosida when he published his classic workThis is a comprehensive book on functional analysis. In Chapters1-5fundamental concepts of functional analysis are introduced, and important concrete functional spaces are illustrated. Naturally these chapters are written in a modern way compared with Banach's book "Théorie des opérations linéaires''. Chapter6deals with the theory of commutative normed rings of the Russian school, which will be a motivation for the following Chapters7-10.

*Functional Analysis*(1965). We say more about this English work below and the reader can read the Preface to each of the six editions as well as extracts from some reviews at THIS LINK.

In 1953 Yosida returned to Osaka University when he was appointed Professor in the Department of Mathematics there. In September 1954 he flew to New York via London, England. His 22 September flight from London to New York was on Pan American Airways. This trip to the United States was made to spend some time at the Institute for Advanced Study at Princeton. He spent the months from September 1954 to January 1955 at the Institute. Back in Japan, Yosida moved to Tokyo when he was appointed as Professor in the Department of Mathematics at the University of Tokyo. He was one of several leading mathematicians who tried to persuade the Japanese government to found a Japanese institute for mathematics. They succeeded but, since Tokyo already had many research institutes, it was decided that the Research Institute for Mathematical Sciences would be part of Kyoto University. Yosida remained a professor at the University of Tokyo until he retired in March 1969 when he reached the age of sixty. In April 1969, he organised an international symposium on functional analysis at the Research Institute for Mathematical Sciences at Kyoto. After retiring from Tokyo University, Yosida became Director of the Research Institute for Mathematical Analysis, Kyoto University, holding this position from 1972 to 1976. After this four year appointment, he became Professor at Gakushuin University, a private university in Tokyo, serving until he reached the age of 70 in 1979.

The International Congress of Mathematicians was held in Edinburgh, Scotland, from 14 to 21 August 1958. Yosida attended the Congress and gave a short presentation entitled

*On the differentiability of a semi-group of operators*. The International Congress of Mathematicians was held in Kyoto, Japan, from 21 to 29 August 1990. Yosida was an Honorary Member of the Organising Committee of the Congress but sadly died in June 1990, two months before the Congress took place.

Yosida published a number of important books. These include

*Lectures on differential and integral equations*(1960),

*Theory of distributions*(Japanese) (1960),

*Functional analysis*(1965), which ran to six editions, and

*Operational Calculus - A Theory of Hyperfunctions*(1984). For extracts from reviews of these books see THIS LINK.

He also published Japanese books including

*Linear Operators*(1943),

*Spectral Analysis*(1947),

*Ergodic Theorems*(1948),

*Introduction to Physical Mathematics*(1949),

*Integral equation theory*(1950),

*Hilbert Space Theory*(1953),

*Methods for solving differential equations*(1954), and

*Measure and Integration theory*(1976).

In addition to these books, Yosida wrote, in collaboration with colleagues, a number of mathematical textbooks in Japanese. These include

*Applied mathematics manual*(1954),

*Basics of Phase Analysis*(1960),

*Applied Mathematics, University Seminar No. 1*(1961),

*Function Analysis and Differential Equations*(1976), and I

*ntroduction to Modern Analysis*(1991).

After retiring from Gakushuin University, he continued to undertake research [1]:-

In fact he tried to popularise this approach by writing an elementary approach with the textbookAfter1980he was mainly interested in modernizing operational calculus from Mikusinski's viewpoint.

*Operational Calculus - A Theory of Hyperfunctions.*H G Dales and D G Crighton write in [3]:-

Yosida played a major role in the Mathematical Society of Japan. He served the Society as its president for a total of seven terms (1957, 1959, 1962, 1964, 1965, 1967, 1972). He was a Member of the Science Council of Japan, and a Member of the Executive Committee of the International Mathematical Union. He was elected a member of the Japan Academy on 13 December 1971. In addition he was honoured with a number of awards. He received the Fujiwara Prize in 1964:-It is, of course, a programme of some antiquity to seek to solve differential equations by reducing them to algebraic equations. One method for this is to use Laplace transformations, and a well-known and powerful technique for implementing this method is Mikusinski's operational calculus. The aim of the present work is "to give a simplified exposition as well as an extension of" this calculus.

The Japan Academy awarded Yosida their Emperor's Prize in 1967. He received the Honorary Medal from the University of Liège in 1970; was elected a Foreign Member of the Royal Society of Liège in 1976; became a member of the American Association for the Advancement of Science in 1977; was elected to the American Academy of Arts and Sciences on 11 May 1977, and became an Officier of the Ordre des Palmes Académiques in 1978. This last mentioned award is given by the French Republic to distinguished academics and leading figure in the field of education.In1959, Ginjiro Fujiwara, a Japanese industrialist and politician, turned over much of his private fortune to a charitable foundation, the Fujiwara Foundation of Science. The Foundation awards the Fujiwara Prize to scientists who have made important contributions to the advancement of science and technology.

The book [1] contains many of Yosida's papers. The editor, Kiyosi Ito, writes:-

The publisher Springer-Verlag writes about the book:-Everyone who reads this volume will be able to picture the image of an outstanding mathematician, Kosaku Yosida, who made a tremendous contribution to promote functional analysis in Japan up to the international level.

Kosaku Yosida is world famous as the founder, together with Einar Hille, of the theory of semigroups. These collected papers provide valuable information on how he started and developed functional analysis and operator theoretical methods in the theory of differential equations and probability theory. The book is divided into10sections following the main topics on which Yosida worked. Each section includes comments by a specialist of the fields for the benefit of the reader. An extensive bibliography of all Yosida's papers, both English and Japanese, is included. This edition will prove invaluable to anyone wanting a comprehensive view of Yosida's work.

**Article by:** *J J O'Connor* and *E F Robertson*

**List of References** (5 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

**Cross-references in MacTutor**

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