Spectral radius formula (Gelfand)

Find the norm, the spectral radius, the set of eigenvalues and the spectrum of an operator $A: l_{2} \rightarrow l_{2}$.

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ be a normal operator (for example, self-adjoint or unitary). Prove that

An operator $A \in \mathcal{L}(X)$, where $X$ is a Banach space, is called quasinilpotent if $\lim_{n \rightarrow \infty }\left\| A^{n}\right\|^{\frac{1}{n}} = 0$. Prove that $A$ is quasinilpotent if and only if $\sigma (A) = \left\{ 0\right\}$.

Consider the operator $A \in \mathcal{L}\left(L_{2}(\left[0,1\right])\right)$, which is defined by the formula $(A x)(t) = \int_{0}^{t} x(s) d s, t \in [0,1]$.

Find the spectrum of the Volterra integral operator in $L_{2}(\left[0,1\right])$ with a continuous kernel $K$, i.e., the operator $(A x)(t) = \int_{0}^{t} K(t, s) x(s) d s, t \in [0,1], x \in L_{2}(\left[0,1\right])$, where $K \in C(\left\{ (t, s) \mid 0 \leq s \leq t \leq 1\right\} )$.

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

Let $X = C\left(\left[0, \frac{1}{2}\right]\right)$ and $(A x)(t) = t x\left(t^{2}\right), t \in \left[0, \frac{1}{2}\right], x \in X$. Prove that $r(A) = 0$.

The operator $A$ in $C(\left[0,1\right])$ is defined by the formula

Prove that $r(A) = 1$.

Prove that $r(A B) = r(B A)$ for all $A, B \in \mathcal{L}(X)$.

Prove Theorem 12.1.

Theorem 12.1: Let $X$ be a Banach space. If $A \in \mathcal{B}(X)$, then the limit $\lim_{n \rightarrow \infty }\left\| A^{n}\right\|^{1 / n}$ exists and

where $r(A) = \max_{\lambda \in \sigma (A)}\left|\lambda \right|$ is the spectral radius of the operator $A$ (Definition 12.2).

Find the spectrum, spectral radius, and resolvent of the operator $(A x)(t) = \int_{0}^{t} x(s) d s$ in the spaces:

Find the spectrum, spectral radius, and resolvent of the operator $A: l_{p} \rightarrow l_{p}, p \in [1, \infty )$:

Let $H$ be a Hilbert space, $A$ a normal operator. Prove that $r(A) = \left\| A\right\|$ (see problem 8.78).

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

Prove that the integral operator

is a Volterra operator in the space

Let $A$ be a Volterra operator and $\lambda \in \mathbb {C}$. Prove that the equation $(I-\lambda A) x = y$ in a Banach space $X$ is solvable for every right-hand side and every $\lambda$. Prove that the solution is representable as the series $x = \sum_{k = 0}^{\infty } \lambda^{k} A^{k} y$.

Let $X$ be a complex infinite-dimensional Banach space, $A \in \mathcal{B}(X)$ and $\sigma (A) = \left\{ 0\right\}$. Is it true that $A$ is compact?

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$ and $\left\| A\right\| \in \sigma (A)$. Prove that $\left\| I+A\right\| = 1+\left\| A\right\|$.

Let $H$ be a Hilbert space and let the operator $A \in \mathcal{B}(H)$ be normal. Prove that $\left\| A^{2}\right\| = \left\| A\right\|^{2}$ and, more generally, $\left\| A^{2^{n}}\right\| = \left\| A\right\|^{2^{n}}$ for every natural $n.$¹

If $\mathcal{A}$ is a Banach algebra with identity and $r: \mathcal{A} \to [0, \infty )$ is the spectral radius, show that $r$ is upper semicontinuous. If $a \in \mathcal{A}$ such that $r(a) = 0$, show that $r$ is continuous at $a$.

Verify the statements made in the following example. A measurable function $k: [0,1] \times [0,1] \to \mathbb {C}$ is called a Volterra kernel if $k$ is bounded and $k(x, y) = 0$ when $x < y$. If $1 \le p \le \infty$ and $k$ is a Volterra kernel, define $V_k: L^p(0,1) \to L^p(0,1)$ by

Then $V_k \in \mathcal{B}(L^p)$ and $\left\| V_k\right\| \le \left\| k\right\|_\infty$. If $k, h$ are Volterra kernels and $(hk)(x, y) = \int_0^1 h(x, t) k(t, y) \, dt$, then $hk$ is a Volterra kernel, $\left\| hk\right\|_\infty \le \left\| h\right\|_\infty \left\| k\right\|_\infty$, and $V_{hk} = V_h V_k$. If $k$ is the characteristic function of $\left\{ (x,y) \in [0,1] \times [0,1] : y < x\right\}$, then $V_k$ is the Volterra operator. If $k$ is any Volterra kernel, then $\sigma (V_k) = \{ 0\}$.

(McCabe) If $\mathcal{X}$ is a complex Banach space and $T \in \mathcal{B}(\mathcal{X})$, show that the following statements are equivalent.

Define $A : L^2(\mathbb {R}) \to L^2(\mathbb {R})$ by $(Af)(x) = \exp (-x^2) f(x-1)$ for all $f$ in $L^2(\mathbb {R})$.

For each operator $A$,

in the sense that the indicated limit always exists and has the indicated value.

Express the norm and the spectral radius of a weighted shift in terms of its weights.

For which sequences $\left\{ p_k\right\}$ of positive numbers does there exist an operator $A$ such that $p_k = \left\| A^k\right\|$, $k = 0, 1, 2, \ldots$?

If $p = \left\{ p_n\right\}$ is a sequence of positive numbers such that $\left\{ p_{n+1}/p_n\right\}$ is bounded, what, in terms of $\left\{ p_n\right\}$, is the spectral radius of the shift on $l^2(p)$?

Construct a quasinilpotent operator whose point spectrum is empty.

Is every quasinilpotent operator analytic?

Is the spectrum of a direct sum of operators always the closure of the union of their spectra?

Is it true that to each operator $A$ and to each positive number $\delta$ there corresponds a positive number $\varepsilon$ such that if $\left\| A-B\right\| < \varepsilon$, then $\left|r(A) - r(B)\right| < \delta$? Equivalently: if $A_n \to A$, does it follow that $r(A_n) \to r(A)$?

If $A$ and $B$ are operators such that $\operatorname {spec}(A + nB) = \operatorname {spec} A$ for $n = 0, 1, 2, \ldots$, does it follow that $B$ is quasinilpotent?

If $r(A) \leq 1$, is $A$ similar to a contraction?

A Volterra operator with a bounded kernel is quasinilpotent.

Is every Volterra operator quasinilpotent?

If $A$ is hyponormal, then $\left\| A^n\right\| = \left\| A\right\|^n$ for every positive integer $n$.

Compute the spectral radius of the Volterra operator $A$ on $L^{2}\left({0,1}\right)$ defined by the formula

Let $A$ be a commutative Banach algebra with unit. The radical of $A$ is defined to be the intersection of all maximal ideals of $A$. Prove that the following three statements about an element $x \in A$ are equivalent:

Find an element $x$ in a Banach algebra $A$ (for instance, a bounded linear operator on a Hilbert space) such that $x^{n} \neq 0$ for all $n > 0$, but $\lim_{n \rightarrow 0}\left\| x^{n}\right\|^{1 / n} = 0$.

Suppose $A$ is a commutative Banach algebra with unit, and let $\Delta$ be the set of all complex homomorphisms of $A$, as in Sec. 18.16. Associate with each $x \in A$ a function $\hat{x}$ on $\Delta$ by the formula

$\hat{x}$ is called the Gelfand transform of $x$. Prove that the mapping $x \rightarrow \hat{x}$ is a homomorphism of $A$ onto an algebra $\hat{A}$ of complex functions on $\Delta$, with pointwise multiplication. Under what condition on $A$ is this homomorphism an isomorphism? (See Exercise 12).

Prove that the spectral radius $\rho (x)$ is equal to

Prove that the range of the function $\hat{x}$ is exactly the spectrum $\sigma (x)$.

Define a shift operator $S$ and a multiplication operator $M$ on $\ell^{2}$ by

Put $T = M S$. Show that $T$ is a compact operator which has no eigenvalue and whose spectrum consists of exactly one point. Compute $\left\| T^{n}\right\|$, for $n = 1,2,3, \ldots$, and compute $\lim_{n \rightarrow \infty }\left\| T^{n}\right\|^{1 / n}$.

Use the identity $(x y)^{n} = x(y x)^{n-1} y$ to prove that $x y$ and $y x$ always have the same spectral radius.

Let $S_{R}$ be the right shift, acting on $\ell^{2}$, as in Exercise 2. Let $\left\{ c_{n}\right\}$ be a sequence of complex numbers such that $c_{n} \neq 0$ but $c_{n} \rightarrow 0$ as $n \rightarrow \infty$. Define $M \in \mathscr {B}\left(\ell^{2}\right)$ by

and define $T \in \mathscr {B}\left(\ell^{2}\right)$ by $T = M S_{R}$.

If $x y = y x$ for some $x$ and $y$ in a Banach algebra, prove that

Let $t$ be a large positive number, and define a norm on $\varphi^{2}$ by

Let $A$ be the algebra of all complex 2-by-2 matrices, with the corresponding operator norm:

For $y \in A$, let $y^{*}$ be the conjugate transpose of $y$. Consider a fixed $x \in A$, namely,

Prove the following statements.

Suppose $A$ is a Banach algebra, $m$ is an integer, $m \geq 2, K < \infty$, and

for every $x \in A$. Show that there exist constants $K_{n} < \infty$, for $n = 1,2,3, \ldots$, such that

(This extends Theorem 11.12).

Put $\phi (t) = \exp \left(-t^{2}\right)$. Define $S \in \mathscr {B}\left(L^{2}\right)$, where $L^{2} = L^{2}(R)$, by

so that $\left(S^{2} f\right)(t) = \phi (t) \phi (t-1) f(t-2)$, etc. (Note that $S$ is presented in its polar decomposition $S = P U$).

Find $S^{*}$. Compute that

Conclude that $S$ is one-to-one, that $\mathscr {R}(S)$ is dense in $L^{2}$, and that $\sigma (S) = \left\{ 0\right\}$. Define $T$, with domain $\mathscr {D}(T) = \mathscr {R}(S)$, by

Prove that $\sigma (T)$ is empty.

Let $A$ be a normed algebra over a complete normed field $K$. Prove that if $x \in A$, then the limit exists

Prove that $\lim_{k \rightarrow \infty } \mathcal{A}^{k}=\mathcal{O} \Longleftrightarrow r(\mathcal{A})<1$.