Properties of invertible operators

Let $X, Y$ and $Z$ be Banach spaces, and let the operators $A \in \mathcal{B}(X, Y), B \in \mathcal{B}(Y, Z)$ have bounded inverses. Prove that $B A$ is also invertible. What can be said about the product of an invertible and a non-invertible operator? What can be said about the product of two non-invertible operators?

Let $X, Y$ be Banach spaces, $A \in \mathcal{B}(X, Y), B \in \mathcal{B}(Y, X)$ and let the operators $A B$ and $B A$ be invertible. What can be asserted about the invertibility of the operators $A$ and $B$?

Let $A = A_{1} A_{2} \ldots A_{n}$ ($A, A_{i} \in \mathcal{B}(X)$, where $X$ is a Banach space). Prove that if all the operators $A_{i}$ are invertible, then $A$ is also invertible. Prove that if the operators $A_{i}, i = 1,2, \ldots , n$, commute, then the invertibility of $A$ implies the invertibility of each of the $A_{i}$.

Let $X$ be a Banach space, $C \in \mathcal{B}(X)$ and $\left\| C\right\| < 1$. Prove that the operator $I+C$ is invertible and

Obtain an explicit form for $(I+C)^{-1}$ as a convergent series of operators.

Let $X$ and $Y$ be Banach spaces, $A$ an invertible operator, $B \in \mathcal{B}(X, Y)$ and $\left\| B\right\| < 1 /\left\| A^{-1}\right\|$. Prove that the operator $A+B$ is invertible and

Obtain an explicit form for $(A+B)^{-1}$ as a series.

Let $T_{r}:\left(x_{1}, x_{2}, \ldots \right) \mapsto \left(0, x_{1}, x_{2}, \ldots \right)$ be the right shift operator in the space $l_{2}$. Prove that if $A \in \mathcal{B}\left(l_{2}\right)$ is an arbitrary operator with $\left\| A\right\| < 1$, then the operator $T_{r}+A$ is not invertible ¹.

Let $H$ be a Hilbert space. Prove that the set of operators for which there exists

Find the distance from the operator $A \in \mathcal{B}\left(L_{2}[0,1]\right)$ to the set of non-invertible operators, if

Let $X$ be a normed space, $A \in \mathcal{B}(X)$, and suppose for some complex $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$ we have $I+\lambda_{1} A+\lambda_{2} A^{2}+\ldots \ldots +\lambda_{n} A^{n} = 0$. Prove that $A$ is invertible.

Let $X$ be a Banach space, $A, B \in \mathcal{B}(X)$ and let the operator $I+A B$ be invertible. Prove that the operator $I+B A$ is also invertible.

Let $X$ be a normed space, $A, B \in \mathcal{B}(X)$ and $A B+A+I = B A+A+I = 0$. Prove that the operator $A$ is invertible.

Let $X$ be a normed space, $A \in \mathcal{IB}(X), B \in \mathcal{B}(X)$ and $A B = B A$. Prove that $A^{-1} B = B A^{-1}$.

Let $X$ be a normed space, $A \in \mathcal{I B}(X)$. Define $k: = \left\| A\right\| \cdot \left\| A^{-1}\right\|$ — the condition number of the operator $A$. Suppose that, when solving the equation $A x = y$ (here $y$ is given and $x$ is to be found), an approximate solution $\tilde{x}$ has been found, for which $A \tilde{x} = \tilde{y}$. Prove the estimate for the relative error

It follows from problem 9.1 that invertibility of the operator $A$ implies the estimate $\inf_{\left\| x\right\| = 1}\left\| A x\right\| > 0$. The converse statement is in general false: one must additionally require surjectivity of the operator $A$.

Let $X$ be a Banach space, and $Y$ a normed space; the operator $A \in \mathcal{B}(X, Y)$ and $\inf_{\left\| x\right\|_{X} = 1}\left\| A x\right\|_{Y} = c > 0$. Prove that the image $\operatorname {Im} A$ is closed and $A$ is invertible as an operator from $X$ onto $\operatorname {Im} A$.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$ a self-adjoint uniformly positive operator on $H$, i.e., $A \geqslant c I$, where $c > 0$. Prove that $A$ is invertible.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H), \inf \left\| A x\right\| > 0$ and $\operatorname {Ker} A^{*} = \left\{ 0\right\}$. Prove that $A$ is invertible.

Give an example of Banach spaces $X, Y$ and a sequence of invertible operators $\left\{ A_{n}\right\}_{1}^{\infty } \in \mathcal{B}(X, Y)$, $A_{n} \rightrightarrows A$, such that the operator $A$ is not invertible.

Prove that if $A_{n} \in \mathcal{B}(X, Y), A_{n} \rightrightarrows A$, all $A_{n}$ are invertible and $\left\| A_{n}\right\|^{-1} \leqslant C$, then $A$ is also invertible.

Give an example of a Banach space $X$ and an operator $A \in \mathcal{I B}(X)$ for which there is a sequence of non-invertible bounded operators $\left\{ A_{n}\right\}_{1}^{\infty }$ such that $A_{n} \xrightarrow {s} A$.

Let $A_{n} \xrightarrow {s} A$ in $\mathcal{B}(X)$, where all $A_{n}$ are invertible and the inverse operators are uniformly bounded: $\left\| A_{n}^{-1}\right\| < C$. Is it true that $A$ is invertible:

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{I B}(X, Y)$. Prove that the operator $A^{\prime }$ is also invertible and $\left(A^{\prime }\right)^{-1} = \left(A^{-1}\right)^{\prime }$. Prove that if $X$ and $Y$ are Hilbert spaces, $A \in \mathcal{IB}(X, Y)$, then $A^{*}$ is also invertible and $\left(A^{*}\right)^{-1} = \left(A^{-1}\right)^{*}$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y), A^{\prime } \in \mathcal{I B}\left(Y^{*}, X^{*}\right)$. Prove that the operator $A$ is also invertible and $A^{-1} = \pi_{X}^{-1}\left(\left(A^{\prime }\right)^{-1}\right)^{\prime } \pi_{Y}$, where $\pi_{X}: X \hookrightarrow X^{* *}, \pi_{Y}: Y \hookrightarrow Y^{* *}$ are the canonical embeddings.

Let $H$ be a Hilbert space, $A$ an invertible self-adjoint operator. Prove that the inverse operator $A^{-1}$ is also self-adjoint. Prove that if a self-adjoint invertible operator $A \geqslant 0$, then $A^{-1} \geqslant 0$ as well.

Let $H$ be a Hilbert space, $A$ and $B$ self-adjoint operators with $I \leqslant A \leqslant B$. Prove that the operators $A$ and $B$ are invertible and $B^{-1} \leqslant A^{-1} \leqslant I$.