Convergence of a sequence on the unit sphere of Bahach or Hilbert space

Question

Let $X$ be a Banach or Hilbert space and $A$ be a bounded linear operator on $X$, and fix an element $x \in X$. Then I want to know that are there any good ways or theories to deal with the convergence of the following sequence $\{ y_n\}$: \begin{align} y_n = \frac{A^nx}{||A^nx||}. \end{align} I'm sorry for an ambiguous question. I welcome any kind of comments.

Answer 1

It does not always converge. As a counter example suppose a finite dimensional space $X = \mathbb{R}^2$, and \begin{equation} A = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix},\quad x = \begin{bmatrix}1\\1\end{bmatrix} \end{equation} Of course $A$ is bounded. Then

\begin{equation} y_n = \begin{cases} \displaystyle{\frac{1}{\sqrt{5}}}\begin{bmatrix}2\\-1 \end{bmatrix}, & n=2m+1, \\ \displaystyle{\frac{1}{\sqrt{2}}}\begin{bmatrix}1\\1 \end{bmatrix}, & n=2m. \end{cases} \end{equation}

In finite dimensional spaces, this is indeed the power iteration that converges to the eigenvector corresponding to the largest eigenvalue (in magnitude) of matrix $A$, and the convergence rate is determined by the ratio of first two largest eigenvalues in magnitude, $|\lambda_1 / \lambda_2|$.

I assume more than boundedness is needed for convergence of $y_n$. Perhaps if $A$ is compact, you may show similar argument for rate of convergence based on the singular values.