**Abraham Plessner**'s father was Ezekiel Plessner, a merchant and owner of a textile factory, and his mother was Marie Niekritsch. At this time Lódz, the city of his birth, was the main centre for textile production in the Russian Empire, with many cotton mills. 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. Over 30% of the population was, like Abraham's family, Jewish. Abraham attended secondary school in Lódz from 1909 to 1918. The particular Oberrealschule he attended was set up to educate the children of tradesmen. Because of the political changes brought about by the military situation in World War I, he had lessons in three different languages during his years at the school. From the time he entered in 1909 until 1916 the language of instruction was Russian. However a major battle was fought over Lódz at the end of 1914 and, by early 1915 the city was in the hands of the Germans. In 1916 the language of instruction at the Oberrealschule was changed to German. 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.

In 1919 he entered the University of Giessen and studied under Ludwig Schlesinger and Friedrich Engel. Schlesinger had taught at the universities of Berlin, Bonn and Cluj before taking up a chair at Gissen in 1911, while Engel had moved to Giessen to take up the chair of mathematics in 1913. Schlesinger quickly saw that his young student was very talented so recommended that he follow the traditional pattern for German students at this time and study at a number of different universities. 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. He also attended Courant's seminar and Landau's seminar. Then in the winter seminar of session 1921/22, he studied in at the University of Berlin where Richard von Mises lectured on differential and integral equations and Ludwig Bieberbach on differential geometry. 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. He underwent a knee operation in 1922 which saw him spend seven months in the University hospital. After this he walked with difficulty, usually using a cane as an aid to walking.

Plessner obtained his doctorate from Giessen in 1922 for a thesis on conjugate trigonometrical series entitled *Zur Theorie der konjugierten trigonometrischen Reihen*. Then he worked in Marburg with Kurt Hensel editing Leopold Kronecker's collected works. During his time in Marburg he published a number of important papers containing theorems and concepts that are studied today. His paper *Über die Konvergenz von trigonometrischen Reihen* (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* (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* (1929), Plessner characterised the absolutely continuous measures among the class of Borel measures. Also while at Marburg he published the book *Lebesguesche Integrale und Fouriersche Reihen* written jointly with his former teacher at Giessen, Ludwig Schlesinger. The book which appeared in 1926, was reviewed by Marshall Stone in [5]. Stone writes:-

[

The book]is intended to be an introductory treatment of the theory of Lebesgue integration, and is consequently more restricted and more elementary than the treatises of Hausdorff, Carathéodory, and Hahn. In point of difficulty, it stands between these and de la Vallée Poussin's monograph in the Borel series. The book is divided into six chapters, whose titles are self-explanatory:(1)The fundamental concepts of the theory of sets;(2)The measure of sets of points;(3)Functions of real variables;(4)The Lebesgue integral;(5)Functions of one and two variables; and(6)Fourier series. The last of these is inserted as an illustration of the importance of the Lebesgue theory in the investigation of a classic problem of analysis. The entire book is thorough, accurate, readable, and well documented.

Kurt Schröder writes in a review:-

The book provides, in a clear and concise manner, an overview of the theory of the Lebesgue integral and its application to the theory of Fourier series, so that it can be read without difficulty by a student in their middle semesters.

Returning to Giessen in 1928 as Ludwig Schlesinger's assistant provided Plessner with only a very small income. On 12 February 1929 Plessner's Habilitationsschrift *Ober Summierbarkeit der trigonometrischen Reihen durch arithmetische Mittel *was submitted to the faculty at Giessen. The Faculty report on the Habilitationsschrift was very positive (see [2]):-

Never have we met a student who has acquired such extensive knowledge of mathematical works within a few semesters. ... we became convinced that Plessner will become a brilliant mathematician ... Considering his(financially)insecure position it is astonishing that he had the strength to make deep speculations and we expect ... that he will make further most important contributions.

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* (1929) and *Über konjugierte trigonometrische Reihen* (1935). This last mentioned paper was reviewed by Józef Marcinkiewicz who wrote:-

This is an important theorem which has been attacked by mathematicians for a long time.

On 27 February 1929, Plessner delivered the required lecture to prove his teaching abilities. His lecture *Über neuere Untersuchungen zu den Grundlagen der Mathematik* was declared "satisfactory", although six of the panel of fourteen professors chose to abstain in the vote. 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. He was told by the city officials that he was required to have 20 years continuous residence in Germany to obtain 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. The Jews had been blamed for Germany's defeat in World War I and anti-Semitism in Germany had steadily increased through the 1920s.

Plessner moved to Berlin in June 1929 thinking that, in Berlin, he would be more able to support himself financially. Schlesinger and Engel asked the Rector of the University of Giessen to request the city officials to speed up his application for German citizenship. 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.

In Moscow, he joined the research group of Nikolai Nikolaevich Luzin. 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. It seems that his interest in functional analysis arose when he read Banach's book *Théorie des opérations linéaires* of 1932. He was greatly respected in Moscow: a colleague wrote (see [2]):-

Abrahm Ezechiel Plessner knew so much that it seemed he knew everything. He understood works in any field and every young mathematician tried to tell him of his new results.

His students noted that his lectures were filled with comments like: this is false and this is trivial. They jokingly wrote:-

Paradise in the sense of Plessner is an abstract space in which all theorems are both false and trivial.

He did not find lecturing easy because of his health problems and standing at a blackboard was in itself quite a difficult task for him. However, this was not an easy time to be working in Moscow for quite different reasons. In 1936, Luzin was the victim of a violent political campaign organized by the Soviet authorities through the newspaper *Pravda*. He was accused of anti-Soviet propaganda and sabotage by publishing all his important results abroad and only minor papers in Soviet journals. Certainly from the time Plessner arrived in Moscow, he published in Russian journals. However, for the first few years his papers, although in Russian journals, were written in German. This changed around 1940 and, from that time on his papers were written in Russian.

In 1939 Plessner published two papers, namely *Zur Spektraltheorie maximaler Operatoren* and *Über Funktionen eines maximalen Operators*. These papers deal with a modified spectral theory and a restricted notion of functions for maximal (but not necessarily self-adjoint) operators. He continued this line of research, publishing *Über halbunitäre Operatoren* later in the same year. 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. In 1941 he published the survey *Spectral theory of linear operators I*. (Russian). Although this was a paper rather than a book, since it is over 120 pages long, it might well have been a book. Jacob David Tamarkin writes:-

The present survey is based on a course of lectures delivered by the author in1938-1939at the Moscow University. It contains, together with an introduction and fundamental notions of the theory of Hilbert spaces and linear operators in Hilbert spaces, the foundations of the spectral theory of linear operators(bounded and not bounded)and of extensions of self-adjoint operators. The spectral decomposition is treated by a method analogous to the complex methods of Poincaré-Hellinger, but presenting some new details. The exposition is compact and elegant. The theory of functions of operators is reserved for the second part of the survey.

He did start to write a book *Spectral theory of linear operators *(Russian) on this topic in 1948 but there were many difficulties caused by illness but also for a quite scandalous reason. Life became very hard after he was dismissed from both posts that he held in 1949. Now this might be a little hard to understand but the reasons must be clear and due primarily to Ivan Matveevich Vinogradov. We quote from [3]:-

Vinogradov's and his associates' anti-Semitism went beyond the general Soviet norms of the time. He was striving to make the Institute and the upper echelons of the Soviet mathematics completely "Judenfrei" and, at least within the Institute, came close to achieving this infamous objective.

Plessner's last years were ones of financial hardship and his health, which as we explained above had never been good, became steadily worse. His role in mathematics is however a major one and he must be considered as a founder of the Moscow school of functional analysis. He left a wife, Nina Andreevna, but they had no children. We mentioned the book *Spectral theory of linear operators *(Russian) which he began in 1948. 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. An English translation was published in 1969. The Russian edition was reviewed by Louis de Branges who writes:-

This expository account of the theory of linear transformations in Hilbert space is concerned exclusively with normal operators. A now classical theorem states that a normal operator A admits an invariant subspace corresponding to any given closed subset D of the complex plane, such that the spectrum of the restriction of A to the subspace is contained in D. The subspace can be chosen so that its orthogonal complement is also an invariant subspace of A and so that the spectrum of the restriction of A to the orthogonal complement is contained in the closure of the complement of D. Furthermore, the spectral theorem states that the operator admits an integral representation in terms of these invariant subspaces. The author's treatment of the theorem emphasizes the technical aspects of the integration process. Thus an existence theorem for invariant subspaces does not appear in the first five hundred pages of the book. This space is devoted instead to the definition of a partially ordered linear space, the definition of a positivity-preserving operator, the concept of a Hilbert space as a special case of a Banach space, fundamental properties of partially defined operators, and the definitions of adjoint, inverse, and closure.

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