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Let $E$ be a real Banach space. Let $\mathcal L(E)$ be the space of bounded linear operators on $E$ and $\mathcal K(E)$ its subspace consisting of compact operators. For $T \in \mathcal L(E)$,

  • we denote by $N(T)$ its kernel and by $R(T)$ its range.
  • we denote by $\rho(T)$ its resolvent set, by $\sigma(T)$ its spectrum, and by $EV(T)$ its set of eigenvalues. Then $EV(T) \subset \sigma(T) = \mathbb R \setminus \rho(T)$.
  • for $\lambda \in EV(T)$, the set $N(T-\lambda I)$ is called the eigenspace corresponding to $\lambda$.

I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,

Let $E := \ell^2$ with its norm $|\cdot|_2$. An element $x\in E$ is denoted by $x=(x_1, x_2, \ldots, x_n,\ldots)$. Consider the operators $$ \begin{align*} S_r x &= (0, x_1, x_2, \ldots, x_{n-1}, \ldots), \\ S_{\ell} x &= (x_2, x_3, x_4, \ldots, x_{n+1}, \ldots), \end{align*} $$ respectively called the right shift and left shift.

  1. Determine $\|S_r\|$ and $\|S_{\ell}\|$. Does $S_r$ or $S_{\ell}$ belong to $\mathcal{K}(E)$?
  2. Prove that $EV(S_r) = \emptyset$.
  3. Prove that $\sigma\left(S_r\right)=[-1,1]$.
  4. Prove that $E V (S_{\ell}) = (-1, 1)$. Determine the corresponding eigenspaces.
  5. Prove that $\sigma(S_{\ell})=[-1,1]$.
  6. Determine the adjoints $S_r^*$ or $S_{\ell}^*$.
  7. Prove that for every $\lambda \in(-1,1)$, the spaces $R (S_r-\lambda I)$ and $R (S_{\ell}-\lambda I)$ are closed. Give an explicit representation of these spaces.
  8. Prove that the spaces $R(S_r \pm I)$ and $R(S_{\ell} \pm I)$ are dense and that they are not closed.

There are possibly subtle mistakes that I could not recognize in below attempt of (8). Could you please have a check on it?

Let $\lambda \in \{\pm 1\}$. It follows from (2, 3, 4, 5) that $S_r +\lambda I$ and $S_\ell + \lambda I$ are injective but not surjective. Then $N (S_r +\lambda I) = \{0\}$ and $N (S_\ell +\lambda I) = \{0\}$. By (6), $(S_r +\lambda I)^* = S_\ell +\lambda I$ and $(S_\ell +\lambda I)^* = S_r +\lambda I$. Clearly, $S_r +\lambda I$ and $S_\ell +\lambda I$ are linear operators that are densely defined and whose graphs are closed. Then $N((S_r +\lambda I)^*)^\perp = \overline{R (S_r +\lambda I)}$ and $N((S_\ell +\lambda I)^*)^\perp = \overline{R (S_\ell +\lambda I)}$ by Corollary 2.18 in the same book. The claim then follows.

Akira
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