Invariant linear functionals

On Certain Conditions for the Existence of Invariant Linear Functionals

Almost multiplicative linear functionals and approximate spectrum

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

Invariant Functionals on Speh Representations

We study Sp2n(R)-invariant functionals on the spaces of smooth vectors in Speh representations of GL2n(R). For even n we give expressions for such invariant functionals using an explicit realization of the space of smooth vectors in the Speh representations. Furthermore, we show that the functional we construct is, up to a constant, the unique functional on the Speh representation which is inva...

Symmetries of Linear Functionals

It is shown that a linear functional on a space of functions can be described by G, a group of its symmetries, together with the restriction of to certain G-invariant functions. This simple consequence of invariant theory has long been used, implicitly, in the construction of numerical integration rules. It is the author's hope that, by showing that these ideas have nothing to do with the origi...

On Strong Invariant Coordinate System (SICS) Functionals

In modern multivariate statistical analysis, affine invariance or equivariance for statistical procedures are properties of paramount interest and importance. A statistical procedure lacking such a property can sometimes acquire it if carried out not on the original data but rather on suitably transformed data, in some cases accompanied by a retransformation back to the original coordinate syst...