# Loewner, Charles (Karl)

# Loewner, Charles (Karl)

(*b*. Lany, Bohemia [now Czechoslovakia], 29 May 1893;*d*. Stanford, California, 8 January 1968)

*mathematics*.

Loewner was the son of Sigmund and Jana Loewner. He studied mathematics with G. Pick at the German University of Prague and received the Ph.D. in 1917. From 1917 to 1922 he was an assistant at the German Technical University of Prague; from 1922 to 1928, assistant and *Privatdozent* at the University of Berlin; from 1928 to 1930, extraordinary professor at the University of Cologne; and from 1930 to 1939, full professor at Charles University in Prague, which he left when the Nazis occupied Czechoslovakia. From 1939 to 1944 he was lecturer and assistant professor at the University of Louisville, Kentucky; from 1945 to 1946, associate professor, and from 1946 to 1951, full professor at Syracuse University; and from 1951 until his retirement in 1963, full professor at Stanford University.

Loewner was married to Elizabeth Alexander, who died in 1955; they had one daughter. Of short stature, soft-spoken, modest, shy (but exceedingly kind to his acquaintances), he had a large number of research students, even after his retirement. His knowledge of mathematics was broad and profound, and included significant parts of mathematical physics. His originality was remarkable; he chose as his problems far from fashionable topics.

One idea pervades Loewner’s work from his Ph.D. thesis: applying Lie theory concepts and methods to semigroups, and applying semigroups to unexpected mathematical situations. This led him in 1923 to a sensational result (4): the first significant contribution to the Bieberbach hypothesis. (A *schlicht* function *f(z)* = Σ a_{n}z^{n} in the unit circle with *a*_{0} = 0, a_{1} = 1, has ǀ a_{n} ǀ ≤n—the case *n = 2* was Bieberbach’s and Loewner proved it for *n = 3*; in its totality the problem is still open.) In 1934 Loewner defined *n*th-order real monotonic functions by the property of staying monotonic if extended to *n*th-degree symmetric matrices (7) and characterized ∞th-order monotonic functions as functions which, analytically extended to the upper half-plane, map it into itself. The semigroups of first- and second-order monotonic mappings are infinitesimally generated; this property breaks down for orders greater than 2 (28, 32). The infinitesimally generated closed subsemigroup of monotonic mappings of infinite order is characterized by *schlicht* extensions to the upper half-plane (21). Loewner studied minimal semigroup extensions of Lie groups; for the group of the real projective line there are two: that of monotonic mappings of infinite order, and its inverse (21). In higher dimensions the question becomes significant under a suitable definition of monotony (30). Loewner also studied semigroups in a more geometrical context; deformation theorems for projective and Moebius translations (19), and infinitesimally generated semigroups invariant under the non-Euclidean or Moebius group (19), particularly if finite dimensionality and minimality are requested (22).

Among Loewner’s other papers, many of which deal with physics, one should be mentioned explicitly: his non-Archimedean measure in Hilbert space (8), which despite its startling originality (or rather because of it) has drawn little attention outside the circle of those who know Loewner’s work.

## BIBLIOGRAPHY

Loewner’s works are

(1) “Untersuchungen über die Verzerrung bei konfor-men Abbildungen des Einheitskreises, die durch Funktionen mit nichtverschwindender Ableitung geliefert werden,” in *Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Math-phys. Klasse***69** (1917), 89-106;

(2) “Über Extremumsätze bei der konformen Abbildung des Äusseren des Einheitskreises,” in *Mathematische Zeitschrift*, **3** (1919), 65-77;

(3) “Eine Anwendung des Koebeschen Verzerrungssatzes auf ein Problem der Hydrodynamik,” *ibid*., 78-86, written with P. Frank;

(4) “Untersuchungen über schlichte konforme Abbildungen des Einheitskreises,” in *Mathematische Annalen*, **89** (1923), 103-121;

(5) “Bemerkung zu einem Blaschkeschen Konvergenzsatze,” in *Jahresbericht der Deutschen Mathematikervereinigung*, **32** (1923), 198-200, written with T. Rado;

(6)Chapters 3 and 16 in P. Frank and R. von Mises, eds., *Die Differential- und Integralgleichungen der Mechanik und Physik* (Brunswick, 1925), ch. 3, 119-192, and ch. 16, 685-737;

(7) “Über monotone Matrixfunktionen,” in *Mathematische Zeitschrift*, **38** (1934), 177-216;

(8) “Grundzüge einer lnhaltslehre im Hilbertschen Raume,” in *Annals of Mathematics*, **40** (1939), 816-833;

(9) “A Topological Characterization of a Class of Integral Operators,” *ibid.,***49** (1948), 316-332;

(10) “Some Classes of Functions Defined by Difference or Differential Inequalities,” in *Bulletin of the American Mathematical Society,***56** (1950), 308-319;

(11) *A Transformation Theory of the Partial Differential Equations of Gas Dynamics,* NACA Technical Report no. 2065 (New York, 1950);

(12) “Generation of Solutions of Systems of Partial Differential Equations by Composition of Infinitesimal Baecklund Transformations,” in *Journal d’analyse mathématique,***2** (1952-1953), 219-242;

(13) “On Generation of Solutions of the Biharmonic Equation in the Plane by Conformal Mappings,” in *Pacific Journal of Mathematics,***3** (1953), 417-436;

(14) “Conservation Laws in Compressible Fluid Flow and Associated Mappings,” in *Journal of Rational Mechanics and Analysis,***2** (1953), 537-561;

(15) “Some Bounds for the Critical Free Stream Mach Number of a Compressible Flow Around an Obstacle,” in *Studies in Mathematics and Mechanics Presented to Richard von Mises* (New York, 1954), 177-183;

(16) “On Some Critical Points of Higher Order,” Technical Note no, 2, Air Force Contract AF 18(600)680 (1954);

(17) “On Totally Positive Matrices,” in *Mathematische Zeitschrift,***63** (1955), 338-340;

(18) “Continuous Groups,” mimeographed notes, University of California at Berkeley (1955);

(19) “On Some Transformation Semigroups,” in *Journal of Rational Mechanics and Analysis,***5** (1956), 791-804;

(20) “Advanced Matrix Theory,” mimeographed notes, Stanford University (1957);

(21) “Semigroups of Conformal Mappings,” in *Seminar on Analytic Functions, Institute for Advanced Study,* I (Princeton, N.J., 1957), 278-288;

(22) “On Some Transformation Semigroups Invariant Under Euclidean and non-Euclidean Isometries,” in *Journal of Mathematics and Mechanics,***8** (1959), 393-409;

(23) “A Theorem on the Partial Order Derived From a Certain Transformation Semigroup,” in *Mathematische Zeitschrift,***72** (1959), 53-60;

(24) “On the Conformal Capacity in Space,” in *Journal of Mathematics and Mechanics,***8** (1959), 411-414;

(25) “On Some Compositions of Hadamard Type in Classes of Analytic Functions,” in *Bulletin of the American Mathematical Society,***65** (1959), 284-286, written with E. Netanyahu;

(26) “A Group Theoretical Characterization of Some Manifold Structures,” Technical Report no. 2 (1962);

(27) “On Some Classes of Functions Associated With Exponential Polynomials,” in *Studies in Mathematical Analysis and Related Topics* (Stanford, Calif., 1962), 175-182, written with S. Karlin;

(28) “On Generation of Monotonic Transformations of Higher Order by Infinitesimal Transformations,” in *Journal d’analyse mathématique,***11** (1963), 189-206;

(29) “On Some Classes of Functions Associated With Systems of Curves or Partial Differential Equations of First Order,” in *Outlines of the Joint Soviet-American Symposium on Partial Differential Equations* (Novosibirsk, 1963);

(30) “On Semigroups in Analysis and Geometry,” in *Bulletin of the American Mathematical Society,***70** (1964), 1-15;

(31) “Approximation on an Arc by Polynomials With Restricted Zeros,” in *Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen,* Section A, **67** (1964), 121-128, written with J. Korevaar;

(32) “On Schlicht-monotonic Functions of Higher Order,” in *Journal of Mathematical Analysis and Applications,***14** (1966), 320-326;

(33) “Some Concepts of Parallelism With Respect to a Given Transformation Group,” in *Duke Mathematical Journal,***33** (1966), 151-164;

(34) “Determination of the Critical Exponent of the Green’s Function,” in *Contemporary Problems in the Theory of Analytic Functions* (Moscow, 1965), pp. 184-187;

(35) “On the Difference Between the Geometric and the Arithmetic Mean of *n* Quantities,” in *Advances in Mathematics,***5** (1971), 472-473, written with H. B. Mann; and

(36) *Theory of Continuous Groups,* with notes by H. Flanders and M. H. Protter (Cambridge, Mass., 1971).

Hans Freudenthal

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**Loewner, Charles (Karl)**