On the Systems of Finite Weights on the Algebra of Bounded Operators and Corresponding Translation Invariant Measures

© 2019, Pleiades Publishing, Ltd. We describe the class of translation invariant measures on the algebra ℬ(ℋ) of bounded linear operators on a Hilbert space ℋ and some of its subalgebras. In order to achieve this we apply two steps. First we show that a total minimal system of finite weights on the operator algebra defines a family of rectangles in this algebra through construction of operator intervals. The second step is construction of a translation invariant measure on some subalgebras of algebra ℬ(ℋ) by the family of rectangles. The operator intervals in the Jordan algebra ℬ(ℋ)sa is investigated. We also obtain some new operator inequalities

On the Systems of Finite Weights on the Algebra of Bounded Operators and Corresponding Translation Invariant Measures

© 2019, Pleiades Publishing, Ltd. We describe the class of translation invariant measures on the algebra ℬ(ℋ) of bounded linear operators on a Hilbert space ℋ and some of its subalgebras. In order to achieve this we apply two steps. First we show that a total minimal system of finite weights on the operator algebra defines a family of rectangles in this algebra through construction of operator intervals. The second step is construction of a translation invariant measure on some subalgebras of algebra ℬ(ℋ) by the family of rectangles. The operator intervals in the Jordan algebra ℬ(ℋ)sa is investigated. We also obtain some new operator inequalities

Analogues of Jacobi and Weyl Theorems for Infinite-Dimensional Tori

The generalization of Jacobi and Weyl theorems on the dynamics of oscillators on an infinite dimensional tori is presented. As in the classical theorems by Jacibi and Weyl on the linear flow on a finite dimensional torus conditions are obtained for periodicity, nonwandering, ergodicity and transitivity on an invariant torus of trajectories of an infinite system of oscillators. We study the ergodicity of a measure on an invariant torus with respect to the flow of a system of oscillators.Comment: 13 pages, 0 figure