On an algebra of pseudodifferential operators

In this paper we introduce an algebra of pseudodifferential operators with symbols of finite smoothness, acting invariantly and continuously in an Orlicz space of functions of exponential type. The concept of point spectral radius
$$ \lim_{m\to\infty}\|A^m(D)f\|_{\Phi }^{1/m} $$
is introduced and its existence is proved. Here $f$ is an arbitrary function in this space, $A(D)$ is an arbitrary element of the algebra, and $\|\cdot\|_{\Phi }$ is the Luxemburg norm. This point spectral radius is evaluated as the supremum of the modulus of $A(D)$ on the support of the Fourier transform of $f$. We evaluate the spectral radius of a pseudodifferential operator. As applications, certain non-convex and convex cases of the well-known Paley–Wiener theorem are obtained. We also consider the solvability of pseudodifferential equations.