Dilatations of linear operators

Yu. L. Kudryashov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 66 (2020) no. 2, pp. 209-220

Résumé

The article is devoted to building various dilatations of linear operators. The explicit construction of a unitary dilation of a compression operator is considered. Then the $J$-unitary dilatation of a bounded operator is constructed by means of the operator knot concept of a bounded linear operator. Using the Pavlov method, we construct the self-adjoint dilatation of a bounded dissipative operator. We consider spectral and translational representations of the self-adjoint dilatation of a densely defined dissipative operator with nonempty set of regular points. Using the concept of an operator knot for a bounded operator and the Cayley transform, we introduce an operator knot for a linear operator. By means of this concept, we construct the $J$-self-adjoint dilatation of a densely defined operator with a regular point. We obtain conditions of isomorphism of extraneous dilations and their minimality.