Per Enflo in Mathematics
■ A Banach space with basis constant greater than 1. Ark Mat. 11 (1973) 103-107
■ A counterexample to the approximation problem in Banach spaces, Acta Mathematica (1973) 130, Number 1, 309-317
This work started in 1967-68 and was completed in 1972. The slow progress and the many obstacles on the way is described in my Autobiography. The two problems - the Basis problem and the Approximation problem - had been open for more than 40 years, and many mathematicians had tried to solve them. I learned about them as a student. They were the big problems in Functional Analysis, that everybody wanted to solve.
In his report on the
AMS had its 75 year celebration in 1991 and for that Professor P.Halmos wrote a report on "the progress of mathematics in the last 75 years". He reported on 22 mathematical discoveries and my solution of the Basis problem was one of them. And on IBM:s big poster over the History of Mathematics over the last 1000 years, the solution of the Basis problem is one of the less than 50 discoveries from the last century that is mentioned.
In 1936 Professor S.Mazur promised a live goose for the solution of a problem, that turned out to be equivalent to the Approximation problem. I received the goose at a ceremony in Warsaw, in December 1972.
■ On the invariant subspace problem in Banach spaces, Seminaire Maurey-Schwartz (1975-76)
■ On the invariant subspace problem in Banach spaces; Mittag-Leffler report (1980), Acta Math. Vol. 158, (submitted 1981, appeared 1987): 213-313
This work started in 1970 and was completed with a first manuscript in 1975. In Professor K.Saxe's book Beginning Functional Analysis, this work is referred to as "arguably his most famous contribution, so far". In a comment on 1 and 2, Professor L. Gårding talks about my "formidable ability to find new ways ...".
■ Investigations on Hilbert's fifth problem for non locally compact groups. Dissertation, Stockholm University, 1970
■ Topological groups in which multiplication on one side is differentiable or linear. Math. Scand. 24 (1969) 195-207
■ Uniform structures and square roots in topological groups. Israel J. Math. 8 (1970) 230-272
■ On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat. 8 (1969) 103-105
■ On a problem of Smirnov. Ark. Math. 8 (1969) 107-10
■ (Unpublished) L1 and l1 are not uniformly homeomorphic. The proof of this result uses that the set of metrical midpoints between two points is much larger in L1 than in l1. This midpoint argument has later been used and taken much further by other researchers.
This work was done in the years 1964-70. It is a study of an infinite-dimensional version of Hilbert's fifth problem for topological groups - together with a study on uniform homeomorphisms between Banach spaces. Notions of non-linear type were introduced. They turned out to be useful also in other areas of Mathematics and in Computer Science.
In 1946 I.Segal showed that groups modeled on Euclidean Space with x→xy differentiable for fixed y, are Lie groups. It was at that time the strongest result on Hilbert's fifth problem, in the sense that it was the weakest assumption on the group operations to get a Lie group. By a very different method I showed that this result generalizes to local groups modeled on Banach Spaces, if one assumes local uniform continuity of (x,y)→xy. This assumption is trivially satisfied in the finite-dimensional case and it cannot be removed in the infinite-dimensional case. If one removes differentiability conditions on the group operations and replaces them with weaker conditions - Lipschitz or Hölder or just uniform continuity - one is led to non-linear geometric problems in Banach Spaces. In the 1960:s, there was no theory developed in this direction and so I started by considering commutative groups modeled on Banach Spaces. And until today, there is no theory of non-commutative groups modeled on Banach Spaces, without differentiability assumptions on the group operations.
For a commutative group modeled (locally) on a Banach Space, Hilbert's fifth problem would ask if the group is (locally) isomorphic to the additive group of the Banach Space. A first step in proving that is to establish a linear structure in the group. And a first step in establishing a linear structure is to solve the equation y = x2 in the group. I managed to do that under some Hölder condition on the group operation, by developing a notion of non-linear type in Banach Spaces. And, somewhat later, I had to develop a stronger non-linear notion to solve an embedding problem of Smirnov. Both these notions have turned out to be useful also in other areas of Mathematics and in Computer Science.
Once one has estabished that a commutative group modeled on a Banach Space is in itself a linear space, one has to decide whether this linear space is isomorphic to the Banach Space it is modeled on. I managed to show that, under a local Lipschitz condition on the group operations, and if the group is modeled on a reflexive Banach Space and if locally x2 =0 implies x=0, then the group is locally the additive group of some Banach Space (0 is assumed to be the unit element of the group). And so, one is finally led to the problem: If a Banach Space is Lipschitz equivalent to a reflexive Banach, is it linearly isomorphic to that Banach Space? This problem is still open. In 1969 I managed to prove: If a Banach Space is uniformly homeomorphic to a Hilbert Space, it is linearly isomorphic to the Hilbert Space (and the conclusion holds also if they are just locally Lipschitz equivalent). At that time, it was the first result of that type. Since then, much research has been done in that direction.
An infinite-dimensional version of Hilbert's fifth problem can be seen as a particular case of a very general question : To what extent do topological notions and results in finite dimensions carry over to Lipschitz (or Hölder or uniform continuity) notions and results in infinite dimensions?
■ Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math., 13 (1972), 281-288
This work was done in September 1971. The main result is that the notions uniformly convex, uniformly smooth, uniformly non-square and super-reflexive are all isomorphically equivalent.
The problem whether uniformly convex and uniformly smooth are isomorphically equivalent had been posed 30 years earlier, in 1941, by M.Day. Since the concepts uniformly convex and uniformly smooth had become increasingly important in Functional Analysis, it had become increasingly important to know whether they are isomorphically equivalent. Before me, R.C.James had solved half the problem. I learned about his result, soon after arriving in Berkeley in the end of August 1971. I felt that I had a chance to solve the other half of the problem - with the general strategies that I had developed in my isolation in Sweden. For me it was an unusually short time to spend on a problem - less than a month.
■ Some results concerning Lp(μ)-spaces (with H.P. Rosenthal), J. Funct. Anal. 14(1973), 325-348
■ On the structure of separable lp spaces (1 < p < oo) (with D. Alspach and E. Odell) Studia Math., 60 (1977), 79-90
■ Subspaces of L1 containing L1. (with T. Starbird) Studia Math. 65. 203- 225 (1979)
■ On the "Three Space Problem" (with J. Lindenstrauss and G. Pisier) Math. Scand. 36 (1975), 199-210
■ (A comment) Far-reaching improvements of the original approach by W. Davis and me have given, so far, the best results on the Banach-Mazur distance between finite-dimensional spaces with symmetric or good unconditional basis. By a mistake (neither mine nor Davis') my work with Davis appeared under just his name. N.Tomczak-Jaegermann, who did major work on these problems, has referred to our approach as the Davis-Enflo method.
These are results in the linear isomorphic theory of Banach spaces (mostly from the period 1970-76). Some samples: Lp spaces are primary - i.e. if Lp a direct sum of two spaces, one of the spaces has to be Lp already. If Y and X/Y are Hilbert spaces then X is "almost" Hilbert space - but not necessarily isomorphic to Hilbert space. My insights from the years 1964-69 were used in these papers.
■ Theoremes de point fixe et d'approximation (with B. Beauzamy); Ark. Mat. Vol. 23 no. 1 (1985): 19-34
■ Contractive projections on lp-spaces (with W. J. Davis) London Math.Soc. Lecture Notes Series 137 (1989), 151–161
■ Contractive projections onto subsets of L1(0,1) Analysis at Urbana, Vol. I, London Math. Soc. Lecture Note Ser. 137 (1989), 162–184
■ Contractive projections onto subsets of Lp-spaces, Lecture Notes in. Pure and Applied Mathematics 136 (1992), 79–94
■ Minimal points and contractive projections. Constructive Theory of Functions, Marin Drinov Publishing House (2006), 98-103
These results are mostly from the late 80:s and early 90:s. The concept "minimal point" is an outgrowth of my efforts in 1 and it turned out to be a special case of the concept Pareto optimal point in Economics. It also turned out to be a fruitful concept in several areas of non-linear theory in Banach Spaces. The first paper contains Fixed point results for contractive maps and an Approximation result for Pareto optimal points in Economics.
The other papers contain characterizations of contractive subsets of lp-spaces (with Davis) and - much harder - the same characterizations in Lp-spaces (by myself). Because of the connection to Approximation theory, one can find special cases of these results as results in Approximation theory. The problem whether "optimal" and "contractive" are equivalent for uniformly convex spaces is still open.
■ Estimations de produits de polynomes, (with B. Beauzamy,) J. of Number Theory, 21-3 (1985), pp. 390-412
■ Products of polynomials in many variables (with B. Beauzamy, E. Bombieri and H. Montgomery) J. of Number Theory, Vol. 36 No. 2 (1990) 219-245
■ The largest coefficient in products of polynomials, Lecture Notes in Pure and Appl. Math, 136, (1992) pp. 97-105
■ Polynomials in many variables: real vs complex norms (with R. Aron and B. Beauzamy), J. Approx. Theory 74 (1993) 181–198
This work started in the 1970:s but most of it is from the 1980:s and 1990:s. The three first papers contain results on products of polynomials in many variables and on "polynomials with concentration at low degrees". These concepts and results grew out of my work on the invariant subspace problem, because they were needed there. They have connections to Number Theory, so two of the papers were published in Journal of Number Theory. The idea that there are product results for polynomials which are independent of the number of variables was a surprise to many (including, initially, myself).
Other researchers have later worked on generalizing results for "polynomials of degree n" to "polynomials with some concentration up to degree n".
The general direction to find results for polynomials which are independent of the number of variables has been fruitful and the fourth paper (in J. of Approximation theory) contains such results.
■ Some Problems in the Interface between Number Theory, Harmonic Analysis and Geometry of Euclidean Space, Quaestiones Mathematicae 18 (1995), 309-323
■ Exponential numbers of linear operators in normed spaces, (with V. Gurariy, V.I. Lomonosov, Yu.I. Lyubich), Linear Algebra and its Applications, 219, pp. 225-260 (1995).
I worked for many years, trying to find better methods to estimate Weyl sums (to me, this seems to be an even more important problem for the future than the much more famous Riemann Hypothesis). The first paper contains most of my efforts, which were never successful. I still have a feeling that the approach could work, if one can just do it right. The paper has almost no results but lots of open questions. It was reviewed by Professor Heath-Brown in Math.Reviews. He said that he understood rather little of the paper, but had been struck by one of the results, which generalized earlier known results. But that result was just a "dead end" application of some known methods, so from my point of view it ought to have been the least interesting part of the paper.
The second paper is an outgrowth of some ideas in the first paper. It turned out that they can be used to obtain stability results for certain linear operators.
■ Extremal vectors and invariant subspaces (with Shamim Ansari) Trans. Amer. Math. Soc. 350 (1998), 539-558
■ Extremal vectors for a class of linear operators, Functional Analysis and Economic Theory, Springer-Verlag (1998), 61-64
■ Some results on extremal vectors and invariant subspaces (with Terje Hõim) Proc. Amer. Math. Soc.131 (2003), 379-387
■ Extremal vectors and rectifiability (with Aderaw Fenta) Quaestiones Mathematicae - vol 34(1) (2011), 119-12
I discovered in the mid 1990:s a connection between different types of extremal vectors and invariant subspaces. Examples of such extremal vectors are i.e. best approximate inverses of non-invertible operators. This gave, in particular, a new, short and more constructive proof of Lomonosov's famous result that operators commuting with compact operators have invariant subspaces. The method has been used by several researchers to obtain new results on invariant subspaces. It gives invariant subspaces for large classes of operators. How far it goes is still open.