$\textsf{J}$-selfadjoint dilations of common form: minimality and isomorphism

So-called minimum dilation plays an important role in dilation theory. The analysis of the known results leads to the study problem of the $\textsf{J}$-self-adjusted general dilation at a minimum. This gives rise to the natural problem of isomorphism of two arbitrary $\textsf{J}$-self-adjust minimum $\textsf{S}$ dilations of a linear operator $A$ with a non-empty set of regular points. The common approach to construction of $\textsf{J}$-self-adjoint dilation for linear operator with nonempty regular point set is considered in this article. To construct the dilation, operators conjugated to maximal simple symmetric operators and boundary doubles of these operators were used. The most common cases of construct the dilation are the previously known $\textsf{J}$-self-adjust dilations. Minimum criteria of constructed dilation have been proved, one of which is formulated in terms of simplicity of operators. connected to maximally simple symmetric operators. In addition, the theorem of isomorphism of two arbitrary minimum $\textsf{J}$-self-defined dilatations of the original operator has been proved by means of the obtained minimum criteria and a description of the areas of definition of operators connected to the maximal simple symmetric operators. For linear operator $A$ with nonempty regular point set and dense domain in Hilbert space were proved such theorems:

• Operator $\textsf{S}$ is a $\textsf{J}$-self-adjoint dilation for operator $A$. This dilation called the $\textsf{J}$-self-adjoint dilation of common form. Different private cases of dilation $S$ were considered too. Solved the problem for minimality of $\textsf{J}$-self-adjoint dilation.
• Arbitrary $\textsf{J}$-self-adjoint dilation for operator $A$ of a common form is minimality iff operators $F_{\pm}$ are simple.
• Arbitrary minimal $\textsf{J}_{1}$- and $\textsf{J}_{2}$-self-adjoint dilations for operator $A$ of a common form are isomorphic.