a brief review of the classical IST stated in terms of Hankel operators and further prepare to prove our main results in Section 5. Section 6, the last one, is devoted to discussions of our results and some corollaries which directly follow from them. It also contains some open problems.

Notation and Preliminaries
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We adhere to standard terminology accepted in Analysis. Namely, $\mathbb{R}_{\pm }\equiv[0,\pm \infty )$, $\mathbb{C}$ is the complex plane, $$\mathbb{C}_{\pm }=\left\{ z\in \mathbb{C}:\pm {\operatorname{Im}}z>0\right\} .$$Through the paper the subscript $\pm $ indicates objects (functions, operators, spaces, etc.) somehow related to $\mathbb{R}_{\pm }$ or $\mathbb{C}_{\pm }$. The bar $\overline{z}$ denotes the complex conjugate of $z$.

When appropriate, we write$$y\eqsim x\text{ in place of \ }y=\limfunc{const}\cdot x$$and similarly whenever convenient $$y\lesssim _{a}x\text{ in place of }y\leq C_{a}x$$with some $C_{a}>0$ dependent on a parameter $a$ but independent of $x$. If $C_{a}$ is an absolute constant we then write $y\lesssim x$.  This will help us keep bulky formulas under control.

We use $\left\Vert \cdot \right\Vert _{X}$ to denote the norm in a Banach (Hilbert) space $X$.

We will need the Gevrey classes $G^{\alpha },\alpha >0,$ on $\mathbb{R}$ of all functions $f$:

$$\left\Vert \partial _{x}^{n}f\right\Vert _{L^{\infty }}\lesssim
_{f}Q_{f}^{n}\left( n!\right) ^{1+\alpha },n=0,1,2,...$$

with 