The Kato conjecture on the square root of operator

In the theory of semigroups of operators it arises the following problem: “Is it true that the domain of square root from regular operator is equal to the domain of square root from adjoint operator?”. This problem was formulated by T. Kato in 1961. It was called the Kato square root problem or the Kato square root conjecture. Sufficient conditions for fulfilment of the Kato conjecture were studied by T. Kato, J. Lions, A. Yagi, and others. J. Lions has proved that strongly elliptic differential operators with smooth coefficients and homogeneous Dirichlet conditions on a smooth boundary satisfy the Kato conjecture. In 1972, A. McIntosh has constructed a counterexample of an abstract regular accretive operator that does not satisfy the Kato conjecture. Therefore later the investigations in this field were mainly devoted to determination of new classes of regular accretive operators satisfying the Kato conjecture. For strongly elliptic differential operators with measurable bounded coefficients corresponding result was obtained by P. Auscher, S. Hofman, A. McIntosh, and P. Tchamitchian.
In this talk we consider new classes of regular accretive operators, satisfying the Kato conjecture: strongly elliptic functional-differential operators with the Dirichlet boundary conditions, elliptic differential-difference operators with degeneration, and strongly elliptic differential-difference operators with mixed boundary conditions. It will be established a connection between boundary value problems for corresponding equations and nonlocal elliptic boundary value problems.