Problems to Solve

Prove that the operator $A^{-1}$, which is the inverse to the linear operator $A$, is also linear.

Let $X = \mathbb {C}^{m}$ endowed with any norm. Prove that for a linear operator $A$ : $X \rightarrow X$ the following statements are equivalent:

Let $A \in \mathcal{L}\left(l_{2}\right)$ be defined by the formula $A x = \left(x_{1}, \frac{x_{2}}{2}, \ldots , \frac{x_{n}}{n}, \ldots \right), x \in l_{2}$. Prove that the range $R(A)$ is not a closed set in $l_{2}$.

Let $X$ be a Banach space, $A \in \mathcal{L}(X),\left\| A\right\| < 1$. Prove that the operator $I+A$ is continuously invertible, and find $(I+A)^{-1}$.

Let $X$ be an NLS, $A, B \in \mathcal{L}(X)$ are continuously invertible. Prove that the operator $A B$ is continuously invertible and $(A B)^{-1} = B^{-1} A^{-1}$.

Let $X$ be an NLS. Prove that the operator $A \in \mathcal{L}(X)$ is continuously invertible if and only if $A^{2}$ is continuously invertible.

Denote by $X_{0}$ and $X_{1}$ the space $\left\{ x \in C^{1}(\left[0,1\right]) \mid x(0) = 0\right\}$ endowed with norms $\left\| x\right\|_{0} = \max_{t \in [0,1]}\left|x(t)\right|$ and $\left\| x\right\|_{1} = \max_{t \in [0,1]}\left|x(t)\right|+\max_{t \in [0,1]}\left|x^{\prime }(t)\right|$, respectively.

Let $A_{i}: C(\left[0,1\right]) \rightarrow X_{i},\left(A_{i} x\right)(t) = \int_{0}^{t} x(s) d s, i = 0,1$. Find $A_{i}^{-1}$. Prove that $A_{1}^{-1}$ is continuous, and $A_{0}^{-1}$ is not bounded.

Let $X_{1}$ be the space from problem $8.10^{\circ }$. Prove that each of the following operators $A: X_{1} \rightarrow C(\left[0,1\right])$ is continuously invertible, and find $A^{-1}$.

Let $K(t, s) = f(t) g(s)$, where $f, g \in C(\left[0,1\right])$ and $\int_{0}^{1} f(t) g(t) d t \neq 1$. Prove that the operator $A: C(\left[0,1\right]) \rightarrow C(\left[0,1\right])$, where

is continuously invertible. Find $A^{-1}$.

Let $K(t, s) = f(t) g(s)$, where $f, g \in C(\left[0,1\right])$. Prove that the operator $A: C(\left[0,1\right]) \rightarrow C(\left[0,1\right])$, where $(A x)(t) = x(t)-\int_{0}^{t} K(t, s) x(s) d s, t \in [0,1]$, is continuously invertible. Find $A^{-1}$.

In the space $C(\left[0,1\right])$ the operators $A$ and $B$ are defined by the formulas $(A x)(t) = (t+1) x(t),(B x)(t) = x\left(t^{2}\right), t \in [0,1]$. Find $(A B)^{-1}$ and $(B A)^{-1}$.

Let $X$ be an NLS, $A, B A \in \mathcal{L}(X)$ be continuously invertible. Prove that the operator $B$ is also continuously invertible.

Let $X$ be an NLS, $A, B \in \mathcal{L}(X)$.

Let $X$ be an NLS, $A, B \in \mathcal{L}(X)$.

Let $X$ be an NLS. Prove that if for an operator $A \in \mathcal{L}(X)$ there exists a sequence $\left\{ x_{n}: n \geq 1\right\} \subset X$ such that $\left\| x_{n}\right\| = 1$ and $A x_{n} \rightarrow 0, n \rightarrow \infty$, then $A$ is not continuously invertible.

Prove that the operator $A: C(\left[0,1\right]) \rightarrow C(\left[0,1\right])$, defined by the formula $(A x)(t) = \alpha (t) x(t), \alpha \in C(\left[0,1\right])$, is continuously invertible if and only if $\alpha (t) \neq 0, t \in [0,1]$.

Let the operators $A_{i}: C(\left[0,1\right]) \rightarrow C^{i}(\left[0,1\right]), i = 0,1$, be defined by the formula $\left(A_{i} x\right)(t) = \int_{0}^{t} e^{t-s} x(s) d s$. For each of the cases $i = 0,1$ answer the following questions:

Let $A, B: C(\left[-1,1\right]) \rightarrow C(\left[-1,1\right])$, where $(A x)(t) = x\left(t^{2}\right),(B x)(t) = x\left(t^{3}\right), t \in [-1,1]$. Prove that $A^{-1}$ does not exist, while $B$ is continuously invertible. Find $B^{-1}$.

Let $X_{2} = \left\{ x \in C^{2}(\left[0,1\right]): x(0) = x(1) = 0\right\}$ with a norm $\left\| x\right\|_{2} = \sum_{k = 0}^{2} \max_{t \in [0,1]}\left|x^{(k)}(t)\right|$, The operator $A: X_{2} \rightarrow C(\left[0,1\right])$ is defined by the formula $(A x)(t) = x^{\prime \prime }(t)-4 x(t), t \in [0,1]$. Find $A^{-1}$ and prove that $A$ is continuously invertible.

Let $X_{3} = \left\{ x \in C^{3}(\left[0,1\right]) \quad : \quad x(0) = x^{\prime }(0) = x^{\prime \prime }(0) = 0\right\} ,\left\| x\right\|_{3} = \sum_{k = 0}^{3} \max_{t \in [0,1]}\left|x^{(k)}(t)\right|$, the operator $A: X_{3} \rightarrow C(\left[0,1\right])$ is defined by the formula $(A x)(t) = x^{\prime \prime \prime }(t)-x^{\prime \prime }(t), t \in [0,1]$. Find $A^{-1}$ and prove that $A$ is continuously invertible.

Is the operator $A: l_{2} \rightarrow l_{2}$ continuously invertible if

Let $A: C(\left[0,1\right]) \rightarrow C(\left[0,1\right]),(A x)(t) = \int_{0}^{1}\left(t s+t^{2} s^{2}\right) x(s) d s$. Prove that the operator $A^{-1}$ does not exist.

Let $A: C(\left[-1,1\right]) \rightarrow C(\left[-1,1\right]),(A x)(t) = \frac{1}{2}(x(t)+x(-t))$. Does $A^{-1}$ exist?

Let $X$ be a Banach space with respect to the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$. Assume that $\left\| \cdot \right\|_{2}$ can be estimated by $\left\| \cdot \right\|_{1}$, i.e., there exists $c > 0$ such that $\left\| x\right\|_{2} \leq c\left\| x\right\|_{1}, x \in X$. Prove that the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ are equivalent. Is the completeness of the space in both norms essential?

What conditions on the function $\alpha \in C(\left[a, b\right])$ imply that

Let $\mu (T) < +\infty , M$ be a subspace of $L_{2}(T, \mu )$, and $M \subset L_{\infty }(T, \mu )$. Prove that

Let $\left\{ e_{n}: n \geq 1\right\}$ be a Schauder basis in a Banach space $X$ with a norm $\left\| \cdot \right\|$ (see problem 1.89). For $x = \sum_{i = 1}^{\infty } \alpha_{i}(x) e_{i}$ we put $P_{k} x = \sum_{i = 1}^{k} \alpha_{i}(x) e_{i}, k \geq 1$, and $||| x ||| = \sup_{k \geq 1}\left\| P_{k} x \right\| , x \in X$. Prove that

Let $\left\{ e_{n}: n \geq 1\right\}$ be a Schauder basis in a Banach space $X$.

Let $\left\{ e_{n}: n \geq 1\right\}$ be a Schauder basis in a Banach space $X,\left\| e_{n}\right\| = 1, n \geq 1$, $\sup_{n \geq 1}\left\| P_{n}\right\| = C < +\infty$, where $P_{n} x = \sum_{i = 1}^{n} \alpha_{i}(x) e_{i}$ for $x = \sum_{i = 1}^{\infty } \alpha_{i}(x) e_{i}$. Prove that any sequence $\left\{ v_{n}: n \geq 1\right\} \subset X$ such that $\sum_{n \geq 1}\left\| e_{n}-v_{n}\right\| < \frac{1}{2 C}$ is also a Schauder basis in $X$.