# Per Enflo: Wikis

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Per Enflo
[[Image:Photograph (Kent State University)|225px|alt=]]
Born 1944
Stockholm, Sweden
Residence Kent, Ohio, United States
Fields Functional analysis
Operator theory
Analytic number theory
Institutions University of California, Berkeley
Stanford University
École Polytechnique, Paris
The Royal Institute of Technology, Stockholm
Kent State University
Alma mater Stockholm University
Doctoral students Nilson Bernardes
Miguel Lacruz
Marie Lövblom
Bruce Reznick
Anthony Weston
Known for Approximation problem
Schauder basis
Hilbert's fifth problem (infinite-dimensional)
uniformly convex renorms of superreflexive Banach spaces
embedding metric spaces (unbounded distortion of cube)
"Concentration" of polynomials at low degree
Invariant subspace problem
Influences Joram Lindenstrauss
Laurent Schwartz
Influenced Bernard Beauzamy
Notable awards Mazur's "live goose" for solving "Scottish Book" Problem 153

Per Enflo is a mathematician who has solved fundamental problems in functional analysis. Three of these problems had been open for more than forty years:

In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms.

Per Enflo was born in Stockholm, Sweden in 1944. Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the University of California, Berkeley, Stanford University, École Polytechnique, (Paris) and The Royal Institute of Technology, Stockholm.

Per Enflo is also a concert pianist.

## Enflo's contributions to functional analysis and operator theory

### Hilbert's fifth problem and embeddings

At Stockholm University, Hans Rådström suggested that Per Enflo consider Hilbert's fifth problem in the spirit of functional analysis.[3] In two years, 1969-1970, Enflo published five papers on Hilbert's fifth problem; these papers are collected in Enflo (1970), along with a short summary. Some of the results of the these papers are described in Enflo (1976) and in the last chapter of Benyamini and Lindenstrauss.

#### Applications in computer science

Enflo's techniques have found application in computer science. Algorithm theorists derive approximation algorithms that embed finite metric-spaces into low-dimensional Euclidean spaces with low "distortion" (in Gromov's terminology for the Lipschitz category; c.f. Banach-Mazur distance). Low-dimensional problems have lower computational complexity, of course. More importantly, if the problems embed well in either the Euclidean plane or the three-dimensional Euclidean space, then geometric algorithms become exceptionally fast.

However, such embedding techniques have limitations, as shown by Enflo's (1969) theorem:[4]

For every $m\geq 2$, the Hamming cube Cm cannot be embedded with "distortion D" (or less) into 2m-dimensional Euclidean space if $D < \sqrt{ m }$. Consequently, the optimal embedding is the natural embedding, which realizes {0,1}m as a subspace of m-dimensional Euclidean space.[5]

This theorem, "found by Enflo [1969], is probably the first result showing an unbounded distortion for embeddings into Euclidean spaces. Enflo considered the problem of uniform embeddability among Banach spaces, and the distortion was an auxiliary device in his proof."[6]

### Geometry of Banach spaces

In 1972 Enflo proved that "every super-reflexive Banach space admits an equivalent uniformly convex norm".[7]

### The basis problem and Mazur's goose

At the Scottish Café on 6 November 1936, Stanislaw Mazur posed the "basis problem" of determining whether every Banach space have a Schauder basis, with Mazur promising a "live goose" as a reward: In 1972, Mazur awarded a live goose to Enflo in a ceremony at the Stefan Banach Center in Warsaw; the "goose reward" ceremony was broadcast throughout Poland.

#### The approximation problem (after Grothendieck)

In fact, Enflo's construction also resolved another outstanding problem, the "Approximation Problem", which had been posed and worked on for years by Alexander Grothendieck (just before Grothendieck turned from functional analysis to algebraic geometry). In mathematics, a Banach space is said to have the approximation property, if every compact operator is a limit of finite rank operators. The converse is always true.

Every Hilbert space has the approximation property. Enflo's solution of the basis problem also provided an example of a Banach space that lacked the approximation property. The relation between the approximation problem and the basis problem had been clarified by Grothendieck.[8]

### Invariant subspace problem and polynomials

In functional analysis, one of the most prominent problems was the invariant subspace problem, which required the evaluation of the truth of the following proposition:

Given a complex Banach space H of dimension > 1 and a bounded linear operator T : H → H, then H has a non-trivial closed T-invariant subspace, i.e. there exists a closed linear subspace W of H which is different from {0} and H such that T(W) ⊆ W.

For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.)

Per Enflo proposed a solution to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987[9] Enflo's long "manuscript had a world-wide circulation among mathematicians"[10] and some of its ideas were described in publications besides Enflo (1976).[11] : Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.[9]

In the 1990s, Enflo developed other approaches to the invariant subspace problem, some of which are discussed in a survey article written by Enflo and Victor Lomonosov (2001).

#### Multiplicative inequalities for homogeneous polynomials

An essential idea in Enflo's construction was "concentration of polynomials at low degrees": For all positive integers m and n, there exists C(m,n) > 0 such that for all homogeneous polynomials P and Q of degrees m and n (in k variables), then

$|PQ|\leq C(m,n)|P|\,|Q|$,

where | P | denotes the sum of the absolute values of the coefficients of P. Enflo proved that C(m,n) does not depend on the number of variables k. Enflo's original proof was simplified by Montgomery.[12]

This result was generalized to other norms on the vector space of homogeneous polynomials. Of these norms, the most used has been the Bombieri norm.

##### Bombieri norm

The Bombieri norm is defined in terms of the following scalar product: $\forall \alpha,\beta \in \mathbb{N}^N$ we have $\langle X^\alpha | X^\beta \rangle = 0$ if $\alpha \neq \beta$

$\forall \alpha \in \mathbb{N}^N$ we define $||X^\alpha||^2 = \frac{|\alpha|!}{\alpha!}$,

where we use use the following notation: if $\alpha = (\alpha_1,\dots,\alpha_N) \in \mathbb{N}^N$, we write $|\alpha| = \Sigma_{i=1}^N \alpha_i$ and $\alpha! = \Pi_{i=1}^N (\alpha_i!)$ and $X^\alpha = \Pi_{i=1}^N X_i^{\alpha_i}.$

The most remarkable property of this norm is the Bombieri inequality:

Let P,Q be two homogeneous polynomials respectively of degree $d^\circ(P)$ and $d^\circ(Q)$ with N variables, then, the following inequality holds:

$\frac{d^\circ(P)!d^\circ(Q)!}{(d^\circ(P)+d^\circ(Q))!}||P||^2 \, ||Q||^2 \leq ||P\cdot Q||^2 \leq ||P||^2 \, ||Q||^2.$

In the above statement, the Bombieri inequality is the left-hand side inequality; the right-hand side inequality means that the Bombieri norm is a norm of the algebra of polynomials under multiplication.

The Bombieri inequality implies that the product of two polynomials cannot be arbitrarily small, and this lower-bound is fundamental in applications like polynomial factorization (or in Enflo's construction of an operator without an invariant subspace).

#### Applications

Enflo's idea of "concentration of polynomials at low degrees" has led to important publications in number theory[13] algebraic and Diophantine geometry[14], and polynomial factorization[15].

## Mathematical biology

Per Enflo has also published articles in mathematical biology, on the DNA of Neanderthal and modern humans (with Hawks & Wolpoff) and on zebra mussels in the Great Lakes of the United States (with Heath).

## Piano

Per Enflo is also a concert pianist.

A child prodigy in both music and mathematics, Enflo performed in a piano recital at age 8 and gave a solo recital at age 11. Enflo won the Swedish competition for young pianists at age 11 in 1956; he won the same competition in 1961.[16]

In 1999 Enflo competed in the first annual Van Cliburn Foundation’s International Piano Competition for Outstanding Amateurs.[17] Enflo performs regularly around Kent and in a Mozart series in Columbus, Ohio.

## References

### Notes

1. ^ Per Enflo: A counterexample to the approximation problem in Banach spaces. Acta Mathematica vol. 130, no. 1, Juli 1973
2. ^ *Enflo, Per (1976), "On the invariant subspace problem in Banach spaces", Séminaire Maurey--Schwartz (1975--1976) Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14-15, Centre Math., École Polytech., Palaiseau, pp. 7, MR0473871
3. ^ Rådström had himself published several articles on Hilbert's fifth problem from the point of view of semigroup theory. Rådström was also the (initial) advisor of Martin Ribe, who wrote a thesis on metric linear spaces that need not be locally convex; Ribe also used a few of Enflo's ideas on metric geometry, especially "roundness", in obtaining independent results on uniform and Lipschitz embeddings (Benyamini and Lindenstrauss). This reference also describes results of Enflo and his students on such embeddings.
4. ^ Theorem 14.4.1 in Matoušek.
5. ^ Matoušek 370.
6. ^ Matoušek 372.
7. ^ Beauzamy 1985, page 298.
8. ^ Joram Lindenstrauss and L. Tzafriri.
9. ^ a b Beauzamy 1988; Yadav.
11. ^ For example, Radjavi and Rosenthal (1982).
12. ^ Schmidt, page 257.
13. ^ Montgomery. Schmidt. Beauzamy and Enflo. Beauzamy, Bombieri, Enflo, and Montgomery
14. ^ Bombieri and Gubler
15. ^ Knuth. Beauzamy, Enflo, and Wang.
16. ^ Saxe.
17. ^

### Bibliography

• Enflo, Per. (1970) Investigations on Hilbert’s fifth problem for non locally compact groups (Stockholm University). Enflo's thesis contains reprints of exactly five papers:
• Enflo, Per; 1969a: Topological groups in which multiplication on one side is differentiable or linear. Math. Scand., 24, s. 195-197.
• Per Enflo (1969). "On the nonexistence of uniform homeomorphisms between Lp spaces". Ark. Mat. 8: pp. 103–5.
• Enflo, Per; 1969b: On a problem of Smirnov. Ark. Math., 8, s. 107-109.
• Enflo, Per; 1970a: Uniform structures and square roots in topological groups I. Israel J. Math. 8, pages 230-252.
• Enflo, Per; 1970b: Uniform structures and square roots in topological groups II. Israel J. Math. 8, pages 253—272.
• Enflo, Per. 1976. Uniform homeomorphisms between Banach spaces. Séminaire Maurey-Schwartz (1975—1976), Espaces, Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18, 7 pp. Centre Math., École Polytech., Palaiseau. MR0477709 (57 #17222) [Highlights of papers on Hilbert's fifth problem and on independent results of Martin Ribe, another student of Hans Rådström]
• Per Enflo (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics 13: pp. 281--288.
• Per Enflo: A counterexample to the approximation problem in Banach spaces. Acta Mathematica vol. 130, no. 1, Juli 1973
• Enflo, Per (1976), "On the invariant subspace problem in Banach spaces", Séminaire Maurey--Schwartz (1975--1976) Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14-15, Centre Math., École Polytech., Palaiseau, pp. 7, MR0473871
• Enflo, Per (1987), "On the invariant subspace problem for Banach spaces", Acta Mathematica 158 (3): 213–313, doi:10.1007/BF02392260, MR892591, ISSN 0001-5962
• Beauzamy, Bernard (1985 [1982]). Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland.
• Beauzamy, Bernard (1988). Introduction to Operator Theory and Invariant Subspaces. North Holland.
• Enrico Bombieri and Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P..
• Grothendieck, A.: Produits tensoriels topologiques et espaces nucleaires. Memo. Amer. Math. Soc. 16 (1955).
• Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming. 2 (Third ed.). Reading, Massachusetts: Addison-Wesley. pp. 439--461, 678-691. ISBN 0-201-89684-2.
• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
• Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977. Springer-Verlag.
• Matoušek, Jiří. Lectures on Discrete Geometry. Springer-Verlag, Graduate Texts in Mathematics, 2002, ISBN 9780387953731.
• Heydar Radjavi and Peter Rosenthal (March 1982). "The invariant subspace problem". The Mathematical Intelligencer 4 (1). doi:10.1007/BF03022994.
• Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York. (Contains short biography of Per Enflo)
• Schmidt, Wolfgang M. (1980 [1996 with minor corrections]) Diophantine approximation. Lecture Notes in Mathematics 785. Springer.
• Yadav, B. S. (2005), "The present state and heritages of the invariant subspace problem", Milan Journal of Mathematics 73: 289–316, doi:10.1007/s00032-005-0048-7, MR2175046, ISSN 1424-9286