1,010 research outputs found
Positive Polynomials on Riesz Spaces
We prove some properties of positive polynomial mappings between Riesz
spaces, using finite difference calculus. We establish the polynomial analogue
of the classical result that positive, additive mappings are linear. And we
prove a polynomial version of the Kantorovich extension theorem.Comment: 12 page
Fourier multiplier theorems involving type and cotype
In this paper we develop the theory of Fourier multiplier operators
, for Banach spaces
and , and an operator-valued symbol. The case has been studied
extensively since the 1980's, but far less is known for . In the scalar
setting one can deduce results for from the case . However, in the
vector-valued setting this leads to restrictions both on the smoothness of the
multiplier and on the class of Banach spaces. For example, one often needs that
and are UMD spaces and that satisfies a smoothness condition. We
show that for other geometric conditions on and , such as the
notions of type and cotype, can be used to study Fourier multipliers. Moreover,
we obtain boundedness results for without any smoothness properties of
. Under smoothness conditions the boundedness results can be extrapolated to
other values of and as long as remains
constant.Comment: Revised version, to appear in Journal of Fourier Analysis and
Applications. 31 pages. The results on Besov spaces and the proof of the
extrapolation result have been moved to arXiv:1606.0327
Representation of non-semibounded quadratic forms and orthogonal additivity
In this article we give a representation theorem for non-semibounded
Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint
operator. The main assumptions are closability of the Hermitean quadratic form,
the direct integral structure of the underlying Hilbert space and orthogonal
additivity. We apply this result to several examples, including the position
operator in quantum mechanics and quadratic forms invariant under a unitary
representation of a separable locally compact group. The case of invariance
under a compact group is also discussed in detail
- …