On linear spaces of skew-symmetric matrices of constant rank

Motivated by the study of congruences of lines, we consider linear systems of skew-symmetric matrices of order six and constant rank four. We prove that, up to projective automorphisms, there exist exactly four types of such linear systems of dimension two, giving four families all of the same dimension.     

On generalized strongly nilpotent matrices

By a theorem of Levitzki, we prove that a generalized strongly nilpotent matrix is similar to a upper triangular matrix with zero diagonal by a scalar matrix, which generalizes a recent result by van den Essen and Hubbers. We also show that a special case of a conjecture of Connell and Zweibel is true.     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.     

On certain positive linear operators in polynomial weight spaces

We consider certain modified Szász-Mirakyan operators A _n (f;r) in polynomial weight spaces of functions of one variable and we study approximation properties of these operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the rank of truncated incidence matrices of linear spaces

It is well known that for every finite linear space the number b of lines is greater or equal to the number v of points of the space. In this paper we investigate the relation between the nonnegative integer b - v and suitable configurations of subspaces of a linear space.     

On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces

Let be a simply connected connected real nilpotent Lie group with Lie algebra , a connected closed subgroup of with Lie algebra and satisfying . Let be the unitary character of with differential at the origin. Let be the unitary representation of induced from the character of . We consider the algebra of differential operators invariant under the action of on the bundle with basis associated to these data. We consider the question of the equivalence between the commutativity of and the finite multiplicities of . Corwin and Greenleaf proved that if is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.

     

The non-existence of certain finite linear spaces

We study the existence of finite linear spaces with v points and n ²+n+2 lines, where n ²+1?v?n ²+n+1. For n?3, there is only one such linear space; it has ten points and fourteen lines.     

On the variety of almost commuting nilpotent matrices

Let V be an n-dimensional vector space over an algebraically closed field and $\mathcal{N}Let V be an n-dimensional vector space over an algebraically closed field and N\mathcal{N} the nilcone of nilpotent endomorphisms of V. We study the variety A = {(X, Y, i, j) ? N ×N ×V ×V^* |[X, Y] = ij} \mathcal{A} = \left\{{(X, Y, i, j) \in \mathcal{N} \times \mathcal{N} \times V \times V^{\ast} \vert [X, Y] = ij}\right\} which is closely related to the variety of pairs of nilpotent n × n matrices with commutator of rank at most 1. We describe its irreducible components: two of them correspond to the pairs of commuting matrices, and n − 2 components of smaller dimension corresponding to the pairs of rank one commutator. In our proof we define a map to the zero fiber of the Hilbert scheme of points and study the image and the fibers.     

On the existence of noncompact bounded linear operators between certain Banach spaces

It is proved that if the Banach spaceE has an unconditional basis and ifF is another Banach space, the following two assertions are equivalent: (1) There is a non-compact bounded linear operator fromE intoF′. (2) The space of bounded linear operators fromE intoF′ has a subspace isomorphic toc ₀. This research was supported in part by National Science Foundation Grants GP 12027 and GU 3171. The author thanks the referee for simplifying the proof.     

On the index of certain nilpotent Lie algebras

We introduce a method of calculation of the index of Lie algebras that are factors of the unitriangular Lie algebra with respect to ideals spanned by subsets of root vectors. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Products of nilpotent matrices

We show that any complex singular square matrix T is a product of two nilpotent matrices A and B with rank A = rank B = rank T except when T is a 2×2 nilpotent matrix of rank one.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

Commutators of nilpotent matrices

A matrix of zero trace is, with certain exceptions, the commutator of nilpotent matrices. It may (so far as we know) be necessary to go outside the original field for the entries of the nilpotent matrices.