Cotype and (q, 1)-summing norm in a Banach space

Summary We use modern probabilistic methods to gain a better understanding of what it means that a Banach space fails to be of cotypeq,q>2. In particular, we prove that a Banach space is of cotypeq if and only if the identity map is (q, 1)-summing. (In a previous work, we had shown that this fails forq=2.)Oblatum 2-III-1992Work partially supported by an NSF grant     

The closed ideal of (c o ,p,q)-summing operators

For 1/p+1/q1, we study the closed ideal formed by the (c _o ,p,q)-summing operators. It turns out thatT:XY does not belong to if and only if it factors the mapId:l _p ^*l _q . By localization, we get the ideal that consists of those operatorsT for which all ultrapowersT ^u are contained in . Operators in the complement of are characterized by the property that they factor the mapsId:l _p ^*n l _q ⁿ uniformly. Our main tools are ideal norms.Supported by DFG grant PI 322/1-2     

On the eigenvalues of (p, 2)-summing operators and constants associated with normed spaces

It is shown that the eigenvalues of (q, 2)-absolutely summing operators are q-th power summable for r > q > 2, but in general not q-th power summable. The method of proof also yields a composition formula for (q, 2)-summing operators which implies that a certain power of these operators is nuclear. Inequalities between (q, r)-summing norms are used to derive estimates for the projection constants and the distance to Hilbert spaces of finite dimensional subspaces of type p and cotype q. One also obtains inequalities between different type and cotype constants of finite dimensional spaces.     

Multilinear extensions of absolutely (p;q;r)-summing operators

In this paper we introduce and study a new class containing the class of absolutely summing multilinear mappings, which we call absolutely (p;q ₁,…,q _m ;r)-summing multilinear mappings. We investigate some interesting properties concerning the absolutely (p;q ₁,…,q _m ;r)-summing m-linear mappings defined on Banach spaces. In particular, we prove a kind of Pietsch’s Domination Theorem and a multilinear version of the Factorization Theorem.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

On (q + t, t)-Arcs of Type (0, 2, t)

In this paper we construct an infinite series of (q + t, t)-arcs of type (0, 2, t). We show that this construction includes the Korchmáros-Mazzocca arcs, and we gain new infinite series of examples, too.     

2-Fusion Systems with Components of Type SL2(q)

We study a problem in fusion systems designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. A 2-fusion system is a 2-group S with a family of injective homomorphisms on subgroups of S. Fusion systems arise in the study of modular representations and classifying spaces, and our results have potential ramifications beyond finite group theory. One problem is to determine S or, whenever possible, the entire 2-fusion system from the knowledge of certain subgroups and maps. We consider the case where S contains a subgroup and maps that arise in the Classification with standard component of type SL₂(q).     

Halving PSL(2,q)

We show that PSL(2,q), q 3(mod 4), contains a subset of half the cardinality of PSL(2,q), which is uniformly 2-homogeneous on the projective line.     

Some (17q, 17, 2) and (25q, 25, 3) BIBD Constructions

We give the explicit construction of a regular (17q, 17, 2)-BIBD for any prime power q 17 (mod 32) such that 2 is not a 4th power in GF(q) and the explicit construction of a regular (25q, 25, 3)-BIBD for any prime power q 25 (mod 48) such that and +3 are non-squares in GF(q).     

Generalized quadrangles of order (q,q 2),q even,containingW(q) as a subquadrangle

In this paper we give a characterization of the generalized quadrangleQ(5,q),q even, in terms of ovoids of its subquadrangles.