On the variety of invariant subspaces of a finite-dimensional linear operator

1982 ◽  
Vol 274 (2) ◽  
pp. 721-721 ◽  
Author(s):  
Mark A. Shayman
Keyword(s):  
Linear Operator ◽  
Invariant Subspaces ◽  
Finite Dimensional

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals Conditional Lie-Bäcklund Symmetries and Reductions of the Nonlinear Diffusion Equations with Source

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Junquan Song ◽  
Yujian Ye ◽  
Danda Zhang ◽  
Jun Zhang
Keyword(s):  
Invariant Subspace ◽  
Dynamic Systems ◽  
Nonlinear Diffusion ◽  
Invariant Subspaces ◽  
Higher Order ◽  
Diffusion Equations ◽  
Canonical Forms ◽  
Nonlinear Diffusion Equations ◽  
Symmetry Approach ◽  
Finite Dimensional

Conditional Lie-Bäcklund symmetry approach is used to study the invariant subspace of the nonlinear diffusion equations with sourceut=e−qx(epxP(u)uxm)x+Q(x,u),m≠1. We obtain a complete list of canonical forms for such equations admit multidimensional invariant subspaces determined by higher order conditional Lie-Bäcklund symmetries. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamic systems.

CHAOS IN WAVE INTERACTIONS

2007 ◽  
Vol 17 (01) ◽  
pp. 85-98 ◽  
Author(s):  
Y. CHARLES LI
Keyword(s):  
Stokes Equation ◽  
Invariant Subspaces ◽  
Navier Stokes ◽  
Wave Interactions ◽  
Navier Stokes Equation ◽  
Finite Dimensional ◽  
Galerkin Truncation

Existence of chaos is proved in finite-dimensional invariant subspaces for both two- and three-wave interactions. For a simple Galerkin truncation of the 2D Navier–Stokes equation, existence of chaos is also proved.

Positive and Z-operators on Closed Convex Cones

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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