Ben Mathes

Linear spaces of nilpotent matrices

Abstract We consider several questions on spaces of nilpotent matrices. We present sufficient conditions for triangularizability and give examples of irreducible spaces. We give a necessary and sufficient condition, in terms of the trace, for all linear combinations of a given set of operators to be nilpotent. We also consider the question of the dimension of a space L of nilpotents on F n . In particular, we give a simple new proof of a theorem due to M. Gerstenhaber concerning the maximal dimension of such spaces.

Sums of idempotents

Abstract Let L k denote the set of those n × n matrices expressible as a sum of k idempotent matrices. We study conditions for membership in L k with “small” k. It is shown that the nontrivial cases are those in which the trace t of a matrix A does not exceed 2ρ − 2, where ρ is the rank of A. For A to belong to L k it is sufficient that t (which is necessarily an integer at least equal to ρ) be greater than 2ρ + 1 − k. In certain cases the results are shown to be sharp. For cyclic matrices and, more generally, for those with a low number of blocks in their rational canonical forms, improved results are obtained. Since the number of idempotent summands is often large, the problem of approximating a real or complex matrix by a member of L k is also considered. It is shown, for example, that if the trace of A is an integer t with ρ ⩽ t ⩽ n, then A is in the closure of L 3, while the smallest k with A ϵ L k may be n.

An application of matricial Fibonacci identities to the computation of spectral norms

Among the most intensively studied integer sequences are the Fibonacci and Lucas sequences. Both are instances of second order recurrences [8], both satisfying sk−2+sk−1 = sk for all integers k, but where the fibonacci sequence (fi) begins with f0 = 0 and f1 = 1, the Lucas sequence (li) has l0 = 2 and l1 = 1. Several authors have recently been interested in the singular values of Toeplitz, circulant, and Hankel matrices that are obtained from the Fibonacci and Lucas sequences (see [1], [2], [3], [12], [13], and [14]), where the authors obtain bounds for the “spectral norms”, i.e. the largest singular value. In [1] a formula is given for the exact value of the spectral norms of the Lucas and Fibonacci Hankel matrices. In this paper, we present the exact value for the spectral norms of Toeplitz matrices involving Fibonacci and Lucas numbers. All matrices and vector spaces will be considered as ones over the complex numbers. If A is a complex matrix, we let A∗ denote the adjoint of A. The singular values of a matrix A = (aij) are defined to be the nonzero eigenvalues of |A| ≡ (A∗A) 1 2 , and they are traditionally enumerated in descending order, s1 ≥ s2 ≥ . . . ≥ sk > 0,

On transitive linear semigroups

Abstract This paper deals with semigroups of linear transformations which act transitively on a finite-dimensional vector space. An explicit canonical form is obtained for the semigroups which lack proper transitive left ideals. The class of such semigroups can be considered to be an extention of the class of transitive groups. It contains all minimal transitive (and hence all sharply transitive) semigroups.

Completely bounded transformations of H-operator spaces

Abstract If U is the commutant of a strictly cyclic unilateral weighted shift with a monotonically decreasing weight sequence, then we show that there is a natural isomorphism of the Banach space of bounded linear maps from U into B ( H ) with the Banach space of bounded linear maps of the trace class operators into H , where H is a separable, infinite dimensional Hilbert space. Under this isomorphism, an operator Φ from U into B ( H ) is completely bounded if and only if its image extends to a bounded map of the Hilbert-Schmidt operators into H . The proof shows that if Φ is only completely row bounded, then Φ is in fact completely bounded. The characterization of the completely bounded maps is then used to prove the existence of a family of completely unbounded representations of U into B ( H ).

Tychonoff by way of uniformity

Let (PCC) denote the statement “products of complete uniform spaces are complete. We give a new proof that (PCC) is equivalent to the axiom of choice. We also show how (PCC) provides us with a gentle proof of Tychonoffs theorem. Copyright c

Functions with dense graphs

Summary We describe how to turn an enumeration of the rational numbers into a function with a dense graph.

Substrictly Cyclic Operators

We initiate the study of substrictly cyclic operators and algebras. As an application of this theory, we are able to give a description of the strongly closed ideals in the commutant of the Volterra operator, and quite a bit more.

Operator Semigroups for which Reducibility Implies Decomposability

AbstractA description of the lattice of invariant subspaces is provided for multiplicative semigroups