Math Problems Vault

Problem 108

Let $X$ be a Banach space. Suppose that $A \in \mathcal{B}(X)$ is an invertible operator. Show that

\sigma \left(A^{-1}\right) = \left\{ \lambda ^{-1}: \lambda \neq 0, \lambda \in \sigma (A)\right\} .

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Problem 109

Let $X$ be a Banach space, let $A \in \mathcal{B}(X)$ and $\lambda \in \mathbb {C}$ . Assume that there exists a sequence $(x_{n})$ in $X$ such that

\left\| x_{n}\right\| = 1, \forall n \in \mathbb {N} \text{ and } A x_{n}-\lambda x_{n} \rightarrow 0 \text{ as } n \rightarrow \infty .

Prove that $\lambda \in \sigma (A)$ .

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Problem 110

(a)(b)

Let $(a_{n})$ and $(b_{n})$ be complex sequences such that

\left|a_{n-1}\right| > \left|a_{n}\right| \xrightarrow {(n \rightarrow \infty )} 0 \text{ and }\left|b_{n-1}\right| > \left|b_{n}\right| \xrightarrow {(n \rightarrow \infty )} 0.

Consider the operator $T: \ell^{2} \rightarrow \ell^{2}$ defined by

T x = \left(a_{1} x_{1}, a_{2} x_{2}+b_{1} x_{1}, a_{3} x_{3}+b_{2} x_{2}, \ldots \right), \quad x = \left(x_{j}\right) \in \ell ^{2}.

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(a)

Show that $T$ is compact.

(b)

Find all eigenvalues and eigenvectors of $T$ .

Problem 111

Let $X$ be a Banach space and let $A \in \mathcal{B}(X)$ such that $A^{n} = 0$ for some $n \in \mathbb {N}$ ( $A$ is nilpotent). Find $\sigma (A)$ .

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Problem 112

Let $P \in \mathcal{B}(X)$ be a projection, i.e., a linear operator on $X$ such that $P^{2} = P$ . Construct the resolvent $R(P ; \lambda )$ of $P$

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Problem 113

Let $C_{\mathbb {R}}$ be the space of all continuous and bounded functions $x(t)$ on $\mathbb {R}$ with norm $\left\| x\right\| = \sup_{\mathbb {R}}\left|x(t)\right|$ . On the space $C_{\mathbb {R}}$ we define the operator $A$ by

(A x)(t) = x(t+c)

where $c \in \mathbb {R}$ is a constant. Prove that

\sigma (A) = \left\{ \lambda \in \mathbb {C}:\left|\lambda \right| = 1\right\}

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Problem 114

(a)(b)

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(a)

Let $H$ be a Hilbert space and $a, b \in H$ . Consider the rank-one operator

U: H \rightarrow H, \quad U(x) = \langle x, a\rangle b, x \in H.

Calculate $\left\| U\right\|$ and the spectral radius $r(U)$ . Show that

r(U) = \left\| U\right\| \Leftrightarrow a, b \text{ are linearly independent. }

(b)

Let $X$ be a Banach space, $x^{*} \in X^{*}$ and $y \in X$ . Consider the rank-one operator

V: X \rightarrow X, \quad V(x) = x^{*}(x) y, x \in X.

Calculate $\left\| V\right\|$ and the spectral radius $r(V)$ . Show that

r(V) = \left\| V\right\| \Leftrightarrow \left|x^{*}(y)\right| = \left\| x^{*}\right\| \left\| y\right\| .

Problem 115

Let $k \in C([0,1] \times [0,1])$ be a given function. Consider the operator

B \in \mathcal{B}(C[0,1]) \quad \text{ defined by }(B u)(s) = \int _{0}^{s} k(s, t) u(t) d t

Find $\sigma (B)$ and $r(B)$ .

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Problem 116

Let $X$ be a Banach space and $A, B \in \mathcal{B}(X)$ . Suppose $A B = B A$ . Prove that

r(A+B) \leq r(A)+r(B),

where $r(T)$ is the spectral radius of an operator $T \in \mathcal{B}(X)$ .

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Problem 117

Let $K \subset \mathbb {C}$ be an arbitrary non-empty compact set. Construct an operator $B \in \mathcal{B}\left(\ell^{p}\right), 1 \leq p \leq \infty$ , such that $\sigma (B) = K$ .

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Problem 118

Let $g \in C[0,1]$ be a fixed function and $A \in \mathcal{B}(C[0,1])$ be defined by

(A f)(t) = g(t) f(t), t \in [0,1].

Find $\sigma (A)$ and construct effectively the resolvent $R(A ; \lambda )$ . Find the eigenvalues and eigenvectors of $A$ .

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Problem 119

Let $X$ be a Banach space and $A, B \in \mathcal{B}(X)$ . Show that for any $\lambda \in \rho (A) \cap \rho (B)$ ,

R(B ; \lambda )-R(A ; \lambda ) = R(B ; \lambda )(A-B) R(A ; \lambda )

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Problem 120

Let $k \in C([0,1] \times [0,1])$ be given. Consider the operator $B \in \mathcal{B}(C[0,1])$ defined by

(B u)(s) = \int _{0}^{s} k(s, t) u(t) d t

Find the spectral radius of $B$ . What is the spectrum of $B$ . (Hint: Prove by induction that

\left|\left(B^{n} u\right)(s)\right| \leq \frac{M^{n}}{n!} s^{n}\left\| u\right\| _{\infty }, \forall n \in \mathbb {N},

for some constant $M > 0$ ).

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Problem 121

Determine the spectra of the left and the right shift operators on $\ell^{2}$ :

\begin{aligned} & R\left(x_{1}, x_{2}, x_{3}, \ldots \right) = \left(0, x_{1}, x_{2}, \ldots \right) \\ & L\left(x_{1}, x_{2}, x_{3}, \ldots \right) = \left(x_{2}, x_{3}, x_{4}, \ldots \right). \end{aligned}

Classify them into point, continuous, and residual spectrum.

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