Banach's Theorem on the Inverse Operator. Examples

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$, and let for some sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ of vectors with $\left\| x_{n}\right\| = 1$ the sequence $A x_{n}$ tend to zero. Prove that $A$ is not invertible.

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$. Prove that if the inverse operator $A^{-1} \in \mathcal{B}(Y, X)$ exists, then it is unique.

Consider in the space $X$ the operators $T_{r} x = \left(0, x_{1}, x_{2}, \ldots , x_{n}, \ldots \right)$ and $T_{l} x = \left(x_{2}, x_{3}, \ldots , x_{n}, \ldots \right)$ of shift right and left, respectively. Prove that these operators are not invertible, but the right shift operator has a left inverse, and the left shift operator has a right inverse operator. Find them.

Give an example of an operator for which a left inverse exists, but is not unique. Give an example of an operator for which a right inverse exists, but is not unique.

Let $X$ and $Y$ be Banach spaces, an operator $A \in \mathcal{B}(X, Y)$. Prove that the uniqueness of

Let $X$ and $Y$ be normed spaces, an operator $A \in \mathcal{B}(X, Y)$ have a right inverse $A_{r}^{-1} \in \mathcal{B}(Y, X)$ and a left inverse $A_{l}^{-1} \in \mathcal{B}(Y, X)$. Prove that then $A_{r}^{-1} = A_{l}^{-1} = A^{-1}$.

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$. Prove that if $A$ is invertible, then it is bijective; if a right inverse $A_{r}^{-1} \in \mathcal{B}(Y, X)$ exists, then $A$ is surjective; and if a left inverse $A_{l}^{-1} \in \mathcal{B}(Y, X)$ exists, then $A$ is injective.

Give an example of a bounded injective operator in Banach spaces for which there is no bounded left inverse operator, but there is a right inverse.

Give an example of a bounded surjective operator in Banach spaces for which there is no bounded right inverse operator, but there is a left inverse.

Let $X$ and $Y$ be Banach spaces, an operator $A \in \mathcal{B}(X, Y)$. Prove that $A$ has a left inverse if and only if two conditions hold:

1. $A$ is injective;

2. $\operatorname {Im} A$ is a closed complemented subspace in $Y$.

Let $X$ and $Y$ be Banach spaces, an operator $A \in \mathcal{B}(X, Y)$. Prove that $A$ has a right inverse if and only if two conditions hold:

1. $A$ is surjective;

2. $\operatorname {Ker} A$ is a complemented subspace in $X$.

Give an example showing that in the Banach inverse operator theorem the completeness condition of the space $Y$ is essential.

Show that in the Banach inverse operator theorem the completeness condition of the space $X$ is essential.

Assuming Theorem 9.1 to be true, prove Theorem 9.2, and conversely.

Theorem 9.1 (the Banach inverse operator theorem): let $X$ and $Y$ be Banach spaces and $A \in \mathcal{B}(X, Y)$ a bijective operator. Then $A^{-1} \in \mathcal{B}(Y, X)$.

Theorem 9.2 (closed graph theorem): let $X$ and $Y$ be Banach spaces. An operator $A \in \mathcal{L}(X, Y)$ is bounded if and only if its graph is closed.

Let $X$ be a Banach space, a linear subspace $Y \subset X$ have finite codimension, and $Y = \operatorname {Im} A$, where $A: Z \rightarrow X$ is a bounded operator, $Z$ a Banach space. Prove that $Y$ is closed (compare with Theorem 11.3).

Theorem 11.3: If $X$ and $Y$ are Banach spaces and $A \in \mathcal{B}(X, Y)$ is a Fredholm operator, that is, $\dim \operatorname {Ker} A < \infty$ and $\operatorname {codim} \operatorname {Im} A < \infty$, then $\operatorname {Im} A$ is closed.

Let $X, Y$ and $Z$ be Banach spaces and $J: Y \rightarrow Z$ an injective continuous linear operator. Let $A \in \mathcal{L}(X, Y)$, with the composition $J A \in \mathcal{B}(X, Z)$. Prove that $A \in \mathcal{B}(X, Y)$.

Does there exist an invertible operator $A \in \mathcal{B}\left(c, c_{0}\right)$?

Let $A \in \mathcal{B}\left(L_{2}[0,1]\right)$, with $\operatorname {Im} A \subset C[0,1]$. Prove that $A \in \mathcal{B}\left(L_{2}[0,1], C[0,1]\right)$. Prove that, in general, if $X, Y, Z$ are Banach spaces, $Y$ is continuously embedded in $Z$, and some operator $A$ acts boundedly from $X$ to $Z$ and has the property $\operatorname {Im} A \subset Y$, then $A \in \mathcal{B}(X, Y)$.

Prove that the differentiation operator $A: W_{2}^{1}[0,1] \rightarrow L_{2}[0,1],(A x)(t) = x^{\prime }(t)$ has a closed graph.

Prove that the functional $f: x \mapsto x(0)$, acting in $L_{2}[0,1]$ with $\mathcal{D}(f) = C[0,1]$, has a non-closed graph.

Give an example of Banach spaces $X$ and $Y$ and an operator $A \in \mathcal{L}(X, Y)$ with a non-closed graph.

Let $X$ be a linear space, $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ two norms, and let $X_{i}: = \left(X,\left\| \cdot \right\|_{i}\right)$ be Banach spaces, with $\left\| \cdot \right\|_{1} \prec \left\| \cdot \right\|_{2}$. Prove that the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ are equivalent.

Let $X$ be a Banach space, $X_{0}$ a closed subspace and $X_{1}$ its linear complement. Prove that the projector onto the subspace $X_{0}$ along the subspace $X_{1}$ is bounded if and only if $X_{1}$ is also closed.

Prove that similar operators are invertible or non-invertible simultaneously. Prove that the norms of unitarily equivalent operators coincide. Give an example of similar operators with different norms.

Let $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots \right)$ be the operator of multiplication by the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$, acting in the space $X$. Prove that $A^{-1}$ exists if and only if $\inf \left\{ \left|\lambda_{n}\right|: n \in \mathbb {N}\right\} > 0$. Find $A^{-1}$ and its norm.

Prove that the operator $A x = \left(0, x_{1}, 0, x_{2}, 0, x_{3}, \ldots \right)$ in the space $l_{2}$ has no right inverse. Find $A_{l}^{-1}$. Prove that the operator $B x = \left(x_{1}, x_{2}, x_{4}, x_{8}, \ldots \right)$ in the space $l_{2}$ has no left inverse. Find $B_{r}^{-1}$.

Let $X = C[0,1],(A x)(t) = a(t) x(t)$ be the operator of multiplication by a continuous function. Prove that this operator is invertible if and only if the function $a$ does not vanish on the segment $[0,1]$. Find the inverse operator.

Let $X = L_{p}[0,1], p \in [1, \infty ],(A x)(t) = a(t) x(t)$ be the operator of multiplication by a function $a \in L_{\infty }[0,1]$. Prove that this operator is invertible if and only if $\underset {t \in [0,1]}{\operatorname {ess} \inf }\left|a(t)\right| > 0$ (for the definition of ess inf see problem 6.27). Find the inverse operator.

Consider in the space

Let $A: C^{1}[0,1] \rightarrow C[0,1],(A x)(t) = x^{\prime }(t)$. Prove that it is not invertible. Find a right inverse operator.

Prove that the Hardy operator $(A x)(t) = \frac{1}{t} \int_{0}^{t} x(s) d s$ in the space $L_{p}[0,1], p \in (1, \infty )$, has neither a left nor a right inverse operator.

Find the inverse operator for the operator $A$ in the space $L_{p}[0,1], p \in [1, \infty ]:$

Find the inverse operator for the operator $A$ in the space $C[0,1]:(A x)(t) = x(t)+\int_{0}^{1} e^{t+s} x(s) d s$.

Let $X \subset Y$ be Banach spaces and $\left\| \cdot \right\|_{Y} \prec \left\| \cdot \right\|_{X}$ on $X$ (then the operator $J: X \rightarrow Y, J x = x$, is continuous, i.e., the space $X$ is embedded in $Y$, see Definition 2.2). Under what conditions on the spaces $X$ and $Y$ does the operator $J$ have

Let $\varphi (t)$ be a function continuous on $[0,1]$ mapping the segment $[0,1]$ into itself. Under what conditions on the function $\varphi$ does the change-of-variable operator $(A x)(t) = x(\varphi (t))$ in the space $C[0,1]$ have