Otto Toeplitz
Encyclopedia
Otto Toeplitz was a German
Jewish mathematician
working in functional analysis
.
in 1905. In 1906 Toeplitz arrived to Göttingen University, which was then the world's leading mathematical center, and he remained there for seven years. Mathematics faculty included David Hilbert
, Felix Klein
, and Hermann Minkowski
. Toeplitz joined a group of young people working with Hilbert: Max Born
, Richard Courant
and Ernst Hellinger
, with whom he collaborated for many years afterward. At that time Toeplitz began to rework the theory of linear functional
s and quadratic form
s on n-dimensional spaces for infinite dimensional spaces. He wrote five papers directly related to spectral theory
of operators which Hilbert was developing. During this period he also published a paper on summation processes and discovered the basic ideas of what are now called the Toeplitz operator
s. In 1913 Toeplitz became an extraordinary professor at the University of Kiel
. He was promoted to a professor in 1920.
In 1911, Toeplitz proposed the inscribed square problem
:
This has been established for convex curves
and smooth curves, but the question remains open in general (2007).
Together with Hans Rademacher
, he wrote a classic of popular mathematics Von Zahlen und Figuren, which was first published in 1930 and later translated into English as Enjoyment of mathematics.
Toeplitz was deeply interested in history of mathematics
. In 1929, he cofounded "Quellen und Studien zur Geschichte der Mathematik" with Otto Neugebauer and Julius Stenzel. Beginning in the 1920s, Toeplitz advocated a "genetic method" in teaching of mathematics, which he applied in writing the book Entwicklung der Infinitesimalrechnung ("The Calculus: A Genetic Approach"). The book introduces the subject by giving an idealized historical narrative to motivate the concepts, showing how they developed from classical problems of Greek mathematics
. It was left unfinished, edited by Gottfried Köthe
and posthumously published in German in 1946 (English translation: 1963).
In 1928 Toeplitz succeeded Eduard Study
at Bonn University. In 1933, the Civil Service Law came into effect and professors of Jewish origin were removed from teaching. Initially, Toeplitz was able to retain his position due to an exception for those who had been appointed before 1914, but he was nonetheless dismissed in 1935. In 1939 he emigrated to Palestine and died in Jerusalem from tuberculosis
a year later.
, who was Toeplitz's assistant in Bonn, described their collaboration:
In his own words:
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
Jewish mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
.
Life and work
Toeplitz's father and grandfather were mathematics teachers. Toeplitz studied mathematics in the University of Breslau and was awarded a doctorate in algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
in 1905. In 1906 Toeplitz arrived to Göttingen University, which was then the world's leading mathematical center, and he remained there for seven years. Mathematics faculty included David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
, and Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
. Toeplitz joined a group of young people working with Hilbert: Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
, Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
and Ernst Hellinger
Ernst Hellinger
Ernst David Hellinger was a German mathematician.-Early years:Ernst Hellinger was born on September 30, 1883 in Striegau, Silesia, Germany to Emil and Julie Hellinger. He grew up in Breslau, attended school and graduated from the Gymnasium there in 1902...
, with whom he collaborated for many years afterward. At that time Toeplitz began to rework the theory of linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s and quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s on n-dimensional spaces for infinite dimensional spaces. He wrote five papers directly related to spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
of operators which Hilbert was developing. During this period he also published a paper on summation processes and discovered the basic ideas of what are now called the Toeplitz operator
Toeplitz operator
In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.- Details :Let S1 be the circle, with the standard Lebesgue measure, and L2 be the Hilbert space of square-integrable functions. A bounded measurable function g on S1 defines a...
s. In 1913 Toeplitz became an extraordinary professor at the University of Kiel
University of Kiel
The University of Kiel is a university in the city of Kiel, Germany. It was founded in 1665 as the Academia Holsatorum Chiloniensis by Christian Albert, Duke of Holstein-Gottorp and has approximately 23,000 students today...
. He was promoted to a professor in 1920.
In 1911, Toeplitz proposed the inscribed square problem
Inscribed square problem
The inscribed square problem is an unsolved question in geometry: Does every plane simple curve contain all four vertices of some square? This is known to be true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. Some early...
:
- Does every Jordan curve contain an inscribed squareSquare (geometry)In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
?
This has been established for convex curves
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
and smooth curves, but the question remains open in general (2007).
Together with Hans Rademacher
Hans Rademacher
Hans Adolph Rademacher was a German mathematician, known for work in mathematical analysis and number theory.-Biography:...
, he wrote a classic of popular mathematics Von Zahlen und Figuren, which was first published in 1930 and later translated into English as Enjoyment of mathematics.
Toeplitz was deeply interested in history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
. In 1929, he cofounded "Quellen und Studien zur Geschichte der Mathematik" with Otto Neugebauer and Julius Stenzel. Beginning in the 1920s, Toeplitz advocated a "genetic method" in teaching of mathematics, which he applied in writing the book Entwicklung der Infinitesimalrechnung ("The Calculus: A Genetic Approach"). The book introduces the subject by giving an idealized historical narrative to motivate the concepts, showing how they developed from classical problems of Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
. It was left unfinished, edited by Gottfried Köthe
Gottfried Köthe
Gottfried Maria Hugo Köthe was an Austrian mathematician working in abstract algebra and functional analysis.- Scientific career :...
and posthumously published in German in 1946 (English translation: 1963).
In 1928 Toeplitz succeeded Eduard Study
Eduard Study
Eduard Study was a German mathematician known for work on invariant theory of ternary forms and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.Study was born in Coburg in the Duchy of...
at Bonn University. In 1933, the Civil Service Law came into effect and professors of Jewish origin were removed from teaching. Initially, Toeplitz was able to retain his position due to an exception for those who had been appointed before 1914, but he was nonetheless dismissed in 1935. In 1939 he emigrated to Palestine and died in Jerusalem from tuberculosis
Tuberculosis
Tuberculosis, MTB, or TB is a common, and in many cases lethal, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis. Tuberculosis usually attacks the lungs but can also affect other parts of the body...
a year later.
Quotes
Here is how Gottfried KötheGottfried Köthe
Gottfried Maria Hugo Köthe was an Austrian mathematician working in abstract algebra and functional analysis.- Scientific career :...
, who was Toeplitz's assistant in Bonn, described their collaboration:
- "Otto liked to take walks and talk about scientific questions. I in fact needed a piece of paper and pencil to write everything down. Toeplitz convinced me that the great outline of research comes to light best in dialog. In his protocols of mutual meetings, sometimes it is marked that the results as drafted were found at a walk along the Rhine river ...As we advanced with our joint work, naturally, new questions arose, and we became more daring, setting ourselves higher goals."
In his own words:
- "...Mathematics and mathematical thinking are not only part of a special science, but are also closely connected with our general culture and its historic development of mathematical thinking, a bridge to the so called Arts and Sciences and the seemingly so non-historic exact sciences can be found...Our main purpose is to help build such a bridge. Not for the sake of history but for the genesis of problems, facts and proofs, for the sake of the decisive turning points of that genesis...By going back to the roots of these conceptions, back through the dust of times past, the scars of long use would disappear and they would be reborn to us as creatures full of life." (Toeplitz, 1926).
See also
- Calderón–Toeplitz operator
- Silverman–Toeplitz theoremSilverman–Toeplitz theoremIn mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular...
- Hellinger–Toeplitz theoremHellinger–Toeplitz theoremIn functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded. By definition, an operator A is symmetric if \langle A x | y \rangle = \langle x | A y\rangle for all x, y in the domain of A...
- Toeplitz algebraToeplitz algebraIn operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l2. Taking l2 to be the Hardy space H2, the Toeplitz algebra consists of elements of the formT_f + K\;...
- Toeplitz matrixToeplitz matrixIn linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...
Books
- Hans RademacherHans RademacherHans Adolph Rademacher was a German mathematician, known for work in mathematical analysis and number theory.-Biography:...
and Otto Toeplitz, The enjoyment of mathematics. Selections from mathematics for the amateur (translated by Herbert Zuckerman), Princeton University Press, 1957
- Otto Toeplitz, The calculus: a genetic approach, The University of Chicago Press, 2007