Question

A subspace Y of a normed space X is said to be invariant under a linear operator T:X→X if T(y) ЄY , Let λЄσp(T) (λ belongs to point spectrum), TεB(X,X) , X be a complex Banach space. Show that eigen space of λ is T-invariant.

Answer 1

Question 1. A subspace $Y$ of a normed space $X$ is said to be invariant under a linear operator $T: X \to X$ if $T(y) \in Y$. Let $\lambda \in \sigma_p(T)$ ($\lambda$ belongs to the point spectrum), $T \in B(X, X)$, $X$ be a complex Banach space. Show that the eigenspace of $\lambda$ is $T$-invariant.

Solution. Recall that the point spectrum of $T$ is the set of all eigenvalues of $T$, i.e., all $\lambda \in \mathbb{C}$ such that $Tx = \lambda x$ for some $x \in X$. For the fixed eigenvalue $\lambda$, the set of such vectors $x$ forms a linear subspace of $X$, which is called the eigenspace of $\lambda$.

Let $Y$ be the eigenspace of $\lambda$. Show that $Y$ is $T$-invariant. Take $y \in Y$ and show that $Ty \in Y$. Indeed, $Ty = \lambda y \in Y$, because $Y$ is a subspace of $X$.