Let K be a field and let UTn(K) and Tn(K) denote the groups of all unitriangular and triangular matrices over field K, respectively. In the paper, the lattices of verbal subgroups of these groups are characterized. Consequently the equalities between certain verbal subgroups and their verbal width are determined. The considerations bring a series of verbal subgroups with exactly known finite wi...

We prove some new equivalences of Anderson’s paving conjectures. Among these are a paving conjecture for positive matrices and for strictly upper triangular matrices.

We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable.

A weighted mean matrix, denoted by (N , pn), is a lower triangular matrix with entries pk/Pn, where {pk} is a nonnegative sequence with p0 > 0, and Pn := ∑n k=0 pk. Mishra and Srivastava [1] obtained sufficient conditions on a sequence {pk} and a sequence {λn} for the series ∑ anPnλn/npn to be absolutely summable by the weighted mean matrix (N , pn). Bor [2] extended this result to absolute sum...

Aweighted mean matrix, denoted by N,pn , is a lower triangular matrix with entries pk/Pn, where {pk} is a nonnegative sequence with p0 > 0, and Pn : ∑n k 0 pk. Mishra and Srivastava 1 obtained sufficient conditions on a sequence {pk} and a sequence {λn} for the series ∑ anPnλn/npn to be absolutely summable by the weighted mean matrix N,pn . Recently Savaş and Rhoades 2 established the correspon...

In this paper, we first consider n × n upper-triangular matrices with entries in a given semiring k. Matrices of this form with invertible diagonal entries form a monoid Bn(k). We show that Bn(k) splits as a semidirect product of the monoid of unitriangular matrices Un(k) by the group of diagonal matrices. When the semiring is a field, Bn(k) is actually a group and we recover a well-known resul...

Let G be the group of n×n upper-triangular matrices with elements in a finite field and ones on the diagonal. This paper applies the character theory of Andre, Carter and Yan to analyze a natural random walk based on adding or subtracting a random row from the row above.

Let T be an operator-weighted shift whose weights are 2-by-2 matrices. We say that, given > 0, T is in the -canonical form if each weight is an upper triangular matrix (aij), with 0 ≤ a11, a22 ≤ 1 and a12 6= 0 implies a11, a22 < . We generalize this concept to operator-weighted shifts whose weights are n-by-n matrices and we show that every polynomially bounded weighted shift, whose weights are...