Gilles Pisier
Encyclopedia
Gilles I. Pisier is a Professor of Mathematics at the Pierre and Marie Curie University
and a Distinguished Professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University
. He is known for his contributions to several fields of mathematics, including functional analysis
, probability theory
, harmonic analysis
, and operator theory
. He has also made fundamental contributions to the theory of C*-algebras. Gilles is the younger brother of French actress Marie-France Pisier
.
s", Pisier and Bernard Maurey developed the theory of Rademacher type, following its use in probability theory
by J. Hoffman–Jorgensen and in the characterization of Hilbert space
s among Banach space
s by S. Kwapien. Using probability
in vector space
s, Pisier proved that super-reflexive
Banach spaces can be renormed with the modulus of uniform convexity having "power type". His work (with Per Enflo
and Joram Lindenstrauss
) on the "three–space problem" influenced the work on quasi–normed
spaces by Nigel Kalton.
s. In the 1990s, he solved two long-standing open problems. In the theory of C*-algebras, he solved, jointly with Marius Junge, the problem of the uniqueness of C* -norms on the tensor product
of two copies of B(H), the bounded linear operators on a Hilbert space
H. He and Junge were able to produce two such tensor norms that are nonequivalent. In 1997, he constructed an operator that was polynomially bounded but not similar to a contraction
, answering a famous question of Paul Halmos
.
for this work. He is also a recipient of the Grands Prix de l'Academie des Sciences de Paris in 1992 and the Salem Prize
in 1979.
, harmonic analysis
, and operator theory
. Among them are:
Pierre and Marie Curie University
The Paris VI University , or the Pierre and Marie Curie University , is a university located on the Jussieu Campus in the Latin Quarter of the 5th arrondissement of Paris, France....
and a Distinguished Professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University
Texas A&M University
Texas A&M University is a coeducational public research university located in College Station, Texas . It is the flagship institution of the Texas A&M University System. The sixth-largest university in the United States, A&M's enrollment for Fall 2011 was over 50,000 for the first time in school...
. He is known for his contributions to several fields of mathematics, including 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...
, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, and operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
. He has also made fundamental contributions to the theory of C*-algebras. Gilles is the younger brother of French actress Marie-France Pisier
Marie-France Pisier
Marie-France Pisier was a French actress. She appeared in numerous films of the French New Wave and twice earned the national César Award for Best Supporting Actress.-Life and career:...
.
Geometry of Banach spaces
In the "local theory of Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s", Pisier and Bernard Maurey developed the theory of Rademacher type, following its use in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
by J. Hoffman–Jorgensen and in the characterization of Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s among Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s by S. Kwapien. Using probability
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
in vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, Pisier proved that super-reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach spaces can be renormed with the modulus of uniform convexity having "power type". His work (with Per Enflo
Per Enflo
Per H. Enflo is a mathematician who has solved fundamental problems in functional analysis. Three of these problems had been open for more than forty years:* The basis problem and the approximation problem and later...
and Joram Lindenstrauss
Joram Lindenstrauss
Joram Lindenstrauss is an Israeli mathematician working in functional analysis. He is professor emeritus of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel.-Biography:...
) on the "three–space problem" influenced the work on quasi–normed
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
spaces by Nigel Kalton.
Operator theory
Pisier transformed the area of operator spaceOperator space
In functional analysis, a discipline within mathematics, an operator space is a Banach space "given together with an isometric embedding into the space B of all bounded operators on a Hilbert space H." The category of operator spaces includes operator algebras....
s. In the 1990s, he solved two long-standing open problems. In the theory of C*-algebras, he solved, jointly with Marius Junge, the problem of the uniqueness of C* -norms on the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of two copies of B(H), the bounded linear operators on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H. He and Junge were able to produce two such tensor norms that are nonequivalent. In 1997, he constructed an operator that was polynomially bounded but not similar to a contraction
Contraction
Contraction may refer to:In physiology:* Muscle contraction, one that occurs when a muscle fiber lengthens or shortens** Uterine contraction, contraction of the uterus, such as during childbirth* Contraction, a stage in wound healing...
, answering a famous question of Paul Halmos
Paul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
.
Awards
In 1997, Pisier received the Ostrowski PrizeOstrowski Prize
The Ostrowski Prize is a mathematics award given every other year by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands...
for this work. He is also a recipient of the Grands Prix de l'Academie des Sciences de Paris in 1992 and the Salem Prize
Salem Prize
The Salem Prize, founded by the widow of Raphael Salem, is awarded every year to a young mathematician judged to have done outstanding work in Salem's field of interest, primarily the theory of Fourier series.-Past winners:...
in 1979.
Books
Pisier has authored several books and monographs in the fields of functional analysisFunctional 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...
, harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, and operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
. Among them are:
- "The Volume of Convex Bodies and Banach Space Geometry", Cambridge University PressCambridge University PressCambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
, 2nd ed., 1999. First published in 1989. - "Introduction to Operator Space Theory", Cambridge University Press, 2003.
- "The Operator Hilbert Space OH, Complex Interpolation and Tensor Norms", Amer Mathematical Society, 1996.
- "Factorization of Linear Operators and Geometry of Banach Spaces", Amer Mathematical Society, 1986.
- "Similarity Problems and Completely Bounded Maps", SpringerSpringer Science+Business Media- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
, 2nd ed., 2001. First published in 1995. - "Random Fourier Series with Applications to Harmonic Analysis", with Michael B. Marcus, Princeton University Press, 1981.