**Abraham Plessner** was a Polish born mathematician who is viewed as a founder of the Moscow school of functional analysis.

- The city had grown rapidly with the population of 15,000 in around 1850 growing to a population of 300,000 by the time Abraham was born.
- Abraham attended secondary school in Lódz from 1909 to 1918.
- However, with Polish independence coming at the end of the war, the language of instruction was changed from German to Polish and, for his final year at school, Plessner was taught in Polish.
- The two best places for mathematics were, Schlesinger said, Göttingen and Berlin; someone as talented as Plessner should certainly study at the top places.
- After three semesters at Giessen, Plessner followed Schlesinger's advice and moved on.
- In 1921 Plessner went to the University of Göttingen where, between May and August, he took courses on Dirichlet series and Galois theory by Edmund Landau; algebraic number fields by Emmy Noether; and the calculus of variations by Richard Courant.
- Issai Schur was leading a seminar on algebra which Plessner attended.
- However, even though he was a young man, Plessner's health was not good.
- Plessner obtained his doctorate from Giessen in 1922 for a thesis on conjugate trigonometrical series entitled Zur Theorie der konjugierten trigonometrischen Reihen Ⓣ(On the theory of conjugate trigonometric series).
- His paper Über die Konvergenz von trigonometrischen Reihen Ⓣ(On the convergence of trigonometric series) (1926) contains what today is sometimes known as the Kolmogorov-Seliverstov-Plessner theorem or Plessner's theorem.
- Plessner's theorem states that if a trigonometric series converges everywhere in a set E of positive measure, then its conjugate series converges almost everywhere in E.
- The paper Über das Verhalten analytischer Funktionen am Rande ihres Definitionsbereichs Ⓣ(On the behavior of analytic functions on the edge of their domain) (1927) contains a result which is today also called Plessner's theorem, concerning the boundary behaviour of functions which are meromorphic in the unit disk.
- More precisely, Plessner's theorem states that any holomorphic function on the unit disk partitions the unit circle, modulo a null set, into two disjoint pieces such that at each point of the first piece, has a non-tangential limit, and at each point of the second piece, the cluster set of any Stolz angle is the whole plane.
- This paper also contains a definition of what today is called a 'Plessner point'.
- A Plessner point for a meromorphic function in the unit disc is a point of the unit circle such that in every Stolz angle at the point the cluster set of the function at the point is the whole plane.
- In Eine Kennzeichunung der totalstetigen Funktionen Ⓣ(An identification of absolutely continuous functions) (1929), Plessner characterised the absolutely continuous measures among the class of Borel measures.
- Returning to Giessen in 1928 as Ludwig Schlesinger's assistant provided Plessner with only a very small income.
- On 12 February 1929 Plessner's Habilitation thesis Ober Summierbarkeit der trigonometrischen Reihen durch arithmetische Mittel Ⓣ(Upper summability of trigonometric series by arithmetic mean) was submitted to the faculty at Giessen.
- we became convinced that Plessner will become a brilliant mathematician ...
- The 54-page manuscript was never published although some of the results, without proofs, are given in two further short papers by Plessner, namely Trigonometrische Reihen Ⓣ(Trigonometric series) (1929) and Über konjugierte trigonometrische Reihen Ⓣ(On conjugate trigonometric series) (1935).
- On 27 February 1929, Plessner delivered the required lecture to prove his teaching abilities.
- Despite the fact that his Habilitationsschrift was an outstanding piece of work and he had passed the lecturing requirement, the Senate refused to give its approval since Plessner was a Russian citizen.
- Now there was no rule that required lecturers to be German citizens, yet the Senate voted to only give Plessner his 'venia legendi' if he acquired German citizenship.
- Of course, one has to wonder whether the reluctance of some professors to support a positive evaluation of his lecture and the Senate's requirement that he obtain German citizenship (which they almost certainly knew was nearly impossible) was more to do with the fact that Plessner was Jewish.
- Plessner moved to Berlin in June 1929 thinking that, in Berlin, he would be more able to support himself financially.
- The Rector did make the request to the Giessen city officials but, as Plessner was no longer resident in Giessen, they declared that they would not take any action.
- In January 1930, Schlesinger and Engel appealed to the Rector to request the Senate of the University of Giessen to confer the 'venia legendi' on Plessner without him having German citizenship.
- However, the Senate decided to postpone a decision indefinitely and so Plessner had no choice; he could not get a lectureship in Germany since he was a Russian citizen so he moved to Moscow.
- Although Moscow had provided a research environment where there was much interest in the area of mathematics that Plessner had studied for his thesis, in fact his interests at this time moved to functional analysis and particularly to spectral theory.
- Certainly from the time Plessner arrived in Moscow, he published in Russian journals.
- In 1939 Plessner published two papers, namely Zur Spektraltheorie maximaler Operatoren Ⓣ(On the spectral theory of maximum operators) and Über Funktionen eines maximalen Operators Ⓣ(On functions of a maximum operator).
- Plessner was promoted to professor in 1939 and held posts both at Moscow University and at the Mathematical Institute of the USSR Academy of Sciences.
- Plessner's last years were ones of financial hardship and his health, which as we explained above had never been good, became steadily worse.
- It was still unfinished on his death but Leonid Mikhailovich Abramov and Boris Mikhailovich Makarov completed the book and it was published in 1965, four years after Plessner died.

Born 13 February 1900, Lódz, Russian Empire (now Poland). Died 18 April 1961, Moscow, USSR.

View full biography at MacTutor

Origin Poland

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive