A characterization of compact multipliers

Abstract:

Let G be a compact abelian group and $\varphi$ a complex-valued function defined on the dual $\Gamma$. The main result of this paper is that $\varphi$ is a compact multiplier of type $(p,q),1 \leqq p < \infty$ and $1 \leqq q \leqq \infty$, if and only if it satisfies the following condition: Given $\varepsilon > 0$ there corresponds a finite set $K \subset \Gamma$ such that $|\sum {a_\gamma }{b_\gamma }\varphi (\gamma )| < \varepsilon$ whenever $P = \sum {a_\gamma }\gamma$ and $Q = \sum {b_\gamma }\gamma$ are trigonometric polynomials satisfying ${\left \| P \right \|_p} \leqq 1,{\left \| Q \right \|_{q’}} \leqq 1$ ($q’$ the conjugate index of q) and ${b_\gamma } = 0$ for $\gamma \in K$. Using the above characterization we obtain the following necessary and sufficient condition for $\varphi$ to be the Fourier transform of a continuous complex-valued function on G: Given $\varepsilon > 0$ there corresponds a finite set $K \subset \Gamma$ such that $|\sum {b_\gamma }\varphi (\gamma )| < \varepsilon$ whenever $Q = \sum {b_\gamma }\gamma$ is a trigonometric polynomial satisfying ${\left \| Q \right \|_1} \leqq 1$ and ${b_\gamma } = 0$ for $\gamma \in K$.