Per Enflo

Products of polynomials in many variables

Abstract We study the product of two polynomials in many variables, in several norms, and show that under suitable assumptions this product can be bounded from below independently of the number of variables.

Some results and open questions on spaceability in function spaces

A subset M of a topological vector space X is called lineable (respectively, spaceable) in X if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y subset of M boolean OR {0}. In this article we prove that, for every infinite dimensional closed subspace X of C[0, 1], the set of functions in X having infinitely many zeros in [0, 1] is spaceable in X. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0, 1] or Muntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0, 1], as well as oscillating and annulling properties of subspaces of C[0, 1].

Extremal vectors and invariant subspaces

For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator K some weak limit of the sequence of minimal vectors is noncyclic for al] operators commuting with K and that for any normal operator N, the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with N. Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if T belongs to a c ertain class C of operators, then the sequence of such vectors converges in norm, and that if T belongs to a subclass of C, then the norm limit is cyclic. INTRODUCTION In this paper we will study diSerent types of extremal vectors for operators on Hilbert space and their connection to invariant subspaces. We present a new method to find invariant subspaces, which we feel is more constructive then the known ones and which gives hyperinvariant subspaces for all compact and all normal operators in a unified way. We feel that the method may give invariant subspaces for several other classes of operators, but so far we have only succeeded in proving this under extra assumptions. The second author started this study by considering the best approximate solutions of the equations Tny = x() for the case when T has dense range and xO is not in the range of T. More precisely, he considered the Yn of smallest norm such that {lTnyn-xoll < 6. Using these backward minimal points, Tnyns, he proved Theorem 4. The first author introduced forward minimal points vns such that llvn-xOll < e and ItTnvnll is minimal. This gives even a simpler proof of Theorem 4, and there is a duality between the forward and the backward minimal points as given in Proposition 2. In this paper we also study geometric properties of vn and Tnyn involving scalar products and limit points. These geometric properties have a connection to invariant subspaces. We show that (Tnyn) converges in norm for normal operators. In relation to the norm convergence of the extrerrlal vectors, we introduce a new class of operators, called operators of class 1z. This class generalizes C1-contractions, and we feel that it is of interest in its own right too. Sections 1-2 are due to the second author, 3-4 are due to the first and the appendix is due to both. Received by the editors October 16, 1995. 1991 Mathematics Subject Classification. Primary 4P7A15. Partially supported by NSF grant number 441003. (D)1998 American Mathematical Society

Sufficiency and duality for multiobjective control problems under generalized (B, ρ)-type I functions

In this paper, we introduce the classes of (B, ρ)-type I and generalized (B, ρ)-type I, and derive various sufficient optimality conditions and mixed type duality results for multiobjective control problems under (B, ρ)-type I and generalized (B, ρ)-type I assumptions.

Extremal Vectors for a Class of Linear Operators

We prove a qualitative result characterizing the behavior of backward minimal vectors introduced in [1].

Harnack's theorem for harmonic compact operator-valued functions

Abstract In this paper we show that harmonic compact operator-valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis. We also give an example which shows that an operator-valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator-valued function. We use our harmonic matrix coefficients characterization to establish a Harnacks theorem for an increasing sequence of harmonic compact self-adjoint operator-valued functions and we show that this Harnacks theorem need not hold when the compactness restriction is dropped.

Some results on extremal vectors and invariant subspaces

In 1996 P. Enflo introduced the concept of extremal vectors and their connection to the Invariant Subspace Problem. The study of backward minimal vectors gives a new method of finding invariant subspaces which is more constructive than the previously known methods. In this article we study the properties and behaviour of extremal vectors, give some new formulas related to backward minimal vectors and improve results from papers by Ansari and Enflo (1998) and Enflo (1998).

On sparse languages L such that LL = σ

Abstract A language L ∈ Σ∗ is said to be sparse if L contains a vanishingly small fraction of all possible strings of length n in Σ∗. C. Ponder asked if there exists a sparse language L such that LL = Σ∗. We answer this question in the affirmative. Several different constructions are provided, using ideas from probability theory, fractal geometry, and analytic number theory. We obtain languages that are optimally sparse, up to a constant factor. Finally, we consider the generalization Lj = Σ∗.

Exponential numbers of linear operators in normed spaces

Abstract Let X be a real or complex normed space, A be a linear operator in the space X, and x ϵ X. We put E(X, A, x) = min{l : l>0, ∥Al x∥ ≠ ∥x∥}, or 0 if ∥Ak x∥ = ∥x∥ for all integer k>0. Then let E(X, A) = supx E(X, A, x) and E(X) = supA E(X, A). If dim X ≥ 2 then E(X) ≥ dim X + 1. A space X is called E-finite if E(X) The main results are following. If X is polynomially normed of a degree p, then it is E-finite; moreover, E(X) ≤ Cpn+p−1 (over R), and E(X) ≤ (C p 2 n+p 2−1 ) 2 (over C). If X is Euclidean complex, then n2 − n + 2 ≤ E(X) ≤ n2 − 1 for n ≥ 3; in particular, E(X) = 8 if n = 3. Also, E(X) = 4 if n = 2. If X is Euclidean real, then [ n 2 ] 2 − [ n 2 ] + 2 ≤ E(X) ≤ n(n + 1) 2 , and E(X) = 3 if n = 2. Much more detailed information on E-numbers of individual operators in the complex Euclidean space is obtained. If A is not nilpotent, then E(X, A) ≤ 2ns − s2, where s is the number of nonzero eigenvalues. For any operator A we prove that E(X, A) ≤ n2 − n + t, where t is the number of distinct moduli of nonzero nonunitary eigenvalues. In some cases E-numbers are “small” and can be found exactly. For instance, E(X, A) ≤ 2 if A is normal, and this bound is achieved. The topic is closely connected with some problems related to the number-theoretic trigonometric sums.

Operators with eigenvalues and extreme cases of stability

In the following, we consider some cases where the point spectrum of an operator is either very stable or very unstable with respect to small perturbations of the operator. The main result is about the shift operator on l 2 , whose point spectrum is what we will call strongly stable. We also give some general perturbation results, including a result about the size of the set of operators that have an eigenvalue.