Locally Linearly Dependent Operators

Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the ...

φ-factorable operators and Weyl-Heisenberg frames on LCA groups

Locally Minimal Projections in B(h)

Given an n-tuple {a1, ..., an} of self-adjoint operators on an infinite dimensional Hilbert space H, we say that a projection p in B(H) is locally minimal for {a1, ..., an} if each pajp (for j = 1, ..., n) is a scalar multiple of p. In Theorem 1.8 we show that for any such {a1, ..., an} and any positive integer k there exists a projection p of rank k that is locally minimal for {a1, ..., an}. I...

Isomorphisms between Spaces of Vector-valued Continuous Functions

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Qt and Q2> t n e spaces of continuous real-valued functions C ^ ) and C(Q2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then QQ^X) and C(Q2;X) are isomorphic. The purpose of this paper is to show that this conclusion is ...