Hilbert space description of classical dynamical systems I

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

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Entropy of dynamical systems and perturbations of operators

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700006489 ◽

1991 ◽

Vol 11 (4) ◽

pp. 779-786 ◽

Cited By ~ 4

Author(s):

Dan Voiculescu

Keyword(s):

Hilbert Space ◽

Dynamical Systems ◽

Lower Bound ◽

Hausdorff Measure ◽

Spectral Measure ◽

Von Neumann ◽

Dimensional Hausdorff Measure ◽

Hilbert Space Operators ◽

Neumann Class ◽

Adjoint Operators

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.

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The Ultrametric Hilbert Space Description of Quantum Measurements with Finite Precision

Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models ◽

10.1007/978-94-009-1483-4_4 ◽

1997 ◽

pp. 131-167

Author(s):

Andrei Khrennikov

Keyword(s):

Hilbert Space ◽

Quantum Measurements ◽

Finite Precision ◽

Space Description

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Positive and Z-operators on Closed Convex Cones

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3782 ◽

2018 ◽

Vol 34 ◽

pp. 444-458

Author(s):

Michael Orlitzky

Keyword(s):

Game Theory ◽

Hilbert Space ◽

Dynamical Systems ◽

Linear Operator ◽

Positive Operator ◽

Convex Cone ◽

Real Hilbert Space ◽

Closed Convex Cone ◽

Nonnegative Matrices ◽

Finite Dimensional

Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

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