Convolution integral operators (Q1617984)

The author considers the space $L_2:=L_2(\mathbb{R}^N)$ consisting of the measurable and square-integrable functions on $\mathbb{R}^{N}$. As in the work by \textit{L. Hörmander} [Acta Math. 104, 93--140 (1960; Zbl 0093.11402)], the author considers the space $L_2^2$ being the set of all continuous linear operators in $L_2$ which commute with shifts. Denoting the set of all convolution integral operators in $L_2$ by $I$, the set of all Carleman convolution integral operators in $L_2$ by $C$, the set of all Akhiezer convolution integral operators in $L_2$ by $B$ and all operators in $I$ which can be written as a product of two operators in $C$ by $A$, the author starts by giving a convolution analogue of a lemma on right multiplication which was given by \textit{B. Misra} et al. [Helv. Phys. Acta 36, 963--980 (1963; Zbl 0134.45803)]. Then, in Theorem 1, the author proves that $B^{s}=A$ and gives two corollaries. In Theorem 3, the author proves that $0\in \left \{ \sigma_{c}(T)\cap \sigma_{c}(T^{\ast})\right \}$ provided that $T\in I$ and $k$ is a kernel of $T$. In Theorem 4, the author proves that $C$ is not dense in $I$ in the operator norm if $N=1.$ In Theorem 5, the author proves that for every $\varepsilon >0$, the operator $T$ belonging to $L_2^2$ can be written as a sum of $D$ and $\Gamma$, where $\Gamma \in C$ with $\left \Vert \Gamma \right \Vert <\varepsilon$ and $Df=\sum_{i=1}^{\infty}\lambda_{i}P_{i}f$, $f\in L_2$, under some further conditions.