This paper investigates the structure of general affine subspaces of ( )  L . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a ...

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. Fo...

Recent work has considered the situation of riddling where, when a chaotic attractor lying in an invariant subspace is transversely stable, the basin of the attractor can be riddled with holes that belong to the basin of another attractor. The existence of invariant subspace often relies on certain symmetry of the system, which is, however, a nongeneric property as system defects and small rand...

For a square system of analytic equations, a Newton-invariant subspace is a set which contains the resulting point of a Newton iteration applied to each point in the subspace. For example, if the equations have real coefficients, then the set of real points form a Newtoninvariant subspace. Starting with any point for which Newton’s method quadratically converges to a solution, this article uses...

We introduce a method to prove existence of invariant manifolds and, at the same time to find simple polynomial maps which are conjugated to the dynamics on them. As a first application, we consider the dynamical system given by a Cr map F in a Banach space X close to a fixed point: F (x) = Ax + N(x), A linear, N(0) = 0, DN(0) = 0. We show that if X1 is an invariant subspace of A and A satisfie...

This article develops data-driven methods to identify eigenfunctions of the Koopman operator associated with a dynamical system and subspaces that are invariant under operator. We build on Extended Dynamic Mode Decomposition (EDMD), method finds finite-dimensional approximation span predefined dictionary functions. propose necessary sufficient condition based application EDMD forward backward i...

In this note, we provide a few sufficient conditions for the invariant subspace problem. Introduction An important open problem in operator theory is the invariant subspace problem. Since the problem is solved for all finite dimensional complex vector spaces of dimension at least 2, H denotes a separable Hilbert space whose dimension is infinite. It is enough to think for a contraction T , that...

New bounds on the canonical angles between an invariant subspace of A and an approximating subspace by the differences between Ritz values and the targeted eigenvalues are obtained. From this result, various bounds are readily available to estimate how accurate the Ritz vectors computed from the approximating subspace may be, based on information on approximation accuracies in the Ritz values. ...

The cheap LQ regulator is reinterpreted as an output nulling problem which is a basic problem of the geometric control theory. In fact, solving the LQ regulator problem is equivalent to keep the output of the related Hamiltonian system identically zero. The solution lies on a controlled invariant subspace whose dimension is characterized in terms of the minimal conditioned invariant of the orig...