Spectrum, adjoint and norm of concrete shift operators

Let $X = l_{2}$. The operators $A, B \in \mathcal{L}\left(l_{2}\right)$ are defined as follows:

Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$ and do the same for the operator $B$.

Let $X = L_{2}(\mathbb {R}),(A x)(t) = x(t+1), t \in \mathbb {R}$. Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A)$, and $\sigma_{r}(A)$.

Let $\frac{1}{p}+\frac{1}{q} = 1, A x = \left(x_{2}, x_{3}, \ldots \right), x \in l_{p}, B x = \left(0, x_{1}, x_{2}, \ldots \right), x \in l_{q}$. Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$ and do the same for the operator $B$, if

Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$, the resolvent, and the spectral radius of the operator $A: l_{p} \rightarrow l_{p}, 1 < p < +\infty$, if

Let $s \in \mathbb {R} \backslash \left\{ 0\right\}$ be a fixed number. The operator $A_{s}$ in $L_{2}(\mathbb {R})$ is defined by the formula $\left(A_{s} x\right)(t) = x(t+s), t \in \mathbb {R}$. Prove that $\sigma_{p}\left(A_{s}\right) = \sigma_{r}\left(A_{s}\right) = \varnothing$, and $\sigma \left(A_{s}\right) = \sigma_{c}\left(A_{s}\right) = \left\{ \lambda \in \mathbb {C} \mid \left|\lambda \right| = 1\right\}$.

Prove that the following operators $A: l_{2} \rightarrow l_{2}$ are linear and continuous. Find their norms.

Find the adjoint operator of an operator $A: l_{2} \rightarrow l_{2}$.

Prove that there does not exist an operator $B \in \mathcal{L}\left(l_{2}\right)$ such that $B^{2} = A$ if

Prove that the operators of right and left shift

are bounded in the spaces $l_{p}, p \in [1, \infty ]$, c, and $c_{0}$. Find the norm, image, and kernel of these operators in each of the spaces.

Prove that the shift operator $(A x)(t) = x(t+a)$ is bounded as an operator in the space

Find the adjoint operator $A^{\prime }$ for the operator $A$ in the spaces $l_{p}, p \in [1, \infty )$, and $c_{0}$:

Prove that the operators of left and right shift are partial isometries of the space $l_{2}$.

Verify the statements in Example 5.3: define $S : l^2 \to l^2$ by $S(\alpha_1, \alpha_2, \ldots ) = (0, \alpha_1, \alpha_2, \ldots )$. Then $S$ is an isometry that is not surjective.

Define $V : L^2(0, \infty ) \to L^2(0, \infty )$ by $(Vf)(t) = f(t-1)$ if $t > 1$ and $(Vf)(t) = 0$ if $t \leq 1$. Show that $V$ is an isometry that is not surjective.

Let $S$ be the unilateral shift and compute $SS^{*}$ and $S^{*}S$. Also compute $S^{n} S^{*n}$ and $S^{*n} S^{n}$.

Define $A: l^{2}(\mathbb {Z}) \rightarrow l^{2}(\mathbb {Z})$ by $A(\ldots , \alpha_{-1}, \hat{\alpha }_{0}, \alpha_{1}, \ldots ) = (\ldots , \hat{\alpha }_{-1}, \alpha_{0}, \alpha_{1}, \ldots )$, where $\hat{\ }$ sits above the coefficient in the $0$-place. Find an invariant subspace of $A$ that does not reduce $A$. This operator is called the bilateral shift.

Say that an operator $A$ on $\mathcal{H}$ is irreducible if the only reducing subspaces for $A$ are $(0)$ and $\mathcal{H}$. Prove:

Let $1 \le p < \infty$ and define $S: l^p \to l^p$ by $S(\alpha_1, \alpha_2, \ldots ) = (0, \alpha_1, \alpha_2, \ldots )$. Compute $S^*$.

Let $S$ be the unilateral shift on $\ell^2$, the Hilbert space operator defined by $S(x_1, x_2, x_3, \ldots ) = (0, x_1, x_2, \ldots )$. Show that $S$ is left invertible but not right invertible.

Let $1 \le p \le \infty$ and suppose $0 < \alpha_1 \le \alpha_2 \le \cdots$ such that $r = \lim \alpha_n < \infty$. Define $A: \ell^p \to \ell^p$ by $A(x_1, x_2, \ldots ) = (0, \alpha_1 x_1, \alpha_2 x_2, \ldots )$. Show that $\sigma (A) = \left\{ z \in \mathbb {C} : \left|z\right| \le r\right\}$ and $\sigma_{ap}(A) = \partial \sigma (A)$. If $\left|\lambda \right| < r$, then $\text{ran}(A - \lambda )$ is closed and has codimension $1$. Also, $\sigma_p(A) = \square$.

Let $\mathcal{A} = \mathcal{B}(l^2)$, let $a =$ the unilateral shift on $l^2$, and let $b = a^*$. Show that $\sigma (ab) \neq \sigma (ba)$.

Let $S$ be the unilateral shift on $l^2$. Show that $C^*(S) \supseteq \mathcal{B}_0(l^2)$ and $C^*(S)/\mathcal{B}_0(l^2)$ is abelian. Show that the maximal ideal space of $C^*(S)/\mathcal{B}_0(l^2)$ is homeomorphic to $\partial \mathbb {D}$.

Let $S$ be the unilateral shift on $l^2(\mathbb {N})$ (II.2.10). Examine the sequences $\{ S^k\}$ and $\{ S^{*k}\}$ and their relation to Exercises 3 and 4.

Let $\mathcal{H}$ be a Hilbert space of dimension $\alpha$ and define $S: \mathcal{H}^{(\infty )} \to \mathcal{H}^{(\infty )}$ by $S(h_1, h_2, \dots ) = (0, h_1, h_2, \dots )$. $S$ is called the unilateral shift of multiplicity $\alpha$.

Define an unbounded weighted shift and determine its adjoint.

Let $S$ be the unilateral shift of multiplicity 1 (see Exercise IX.6.4) and find the symmetric operator $A$ such that $S$ is the Cayley transform of $A$.

Let $U = S^*$, where $S$ is the unilateral shift of multiplicity 1. Is $U$ the Cayley transform of a symmetric operator $A$? If so, find it.

Let $S$ be the unilateral shift of multiplicity $1$ on $l^2(\mathbb {N})$ and find $\sigma_l(S)$ and $\sigma_r(S)$.

Does the unilateral shift of multiplicity $1$ have a square root?

If $U$ is the unilateral shift of multiplicity $1$ and $\pi : \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H})$ is the natural map, show that $\pi (U)$ is normal in $\mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H})$ but there is no normal operator $N$ such that $U - N$ is compact. (See Exercise IX.8.14.)

Let $S$ be the operator defined in Example 2.11 where $G = \mathbb {D}$. Show that there is a compact operator $K$ such that $S + K$ is unitarily equivalent to the unilateral shift.

If $U$ and $V$ are isometries, show that there is a path consisting entirely of isometries connecting $U$ to $V$ if and only if $\dim (\mathrm{ran}\ U)^\perp = \dim (\mathrm{ran}\ V)^\perp$.

Let $L, R$ be the left- and right-shift operators on $\ell^{2}$ defined in Exercise 1.64. Show that $R = L^{*}$.

Let $H = \ell^{2}$ be the well known Hilbert space. Consider the left shift defined by

Consider the right-shift on the Hilbert space $\ell^{2}$:

Define its adjoint operator.

Let $S_{R}$ and $S_{L}$ be as above. Prove that neither $S_{R}$ nor $S_{L}$ has polar decompositions $U P$, with $U$ unitary and $P \geq 0$.