General properties of compact operators

Prove that an operator $A \in \mathcal{B}(X, Y)$ is compact if and only if the image of the unit ball in $X$ is a precompact set in $Y$.

Prove that a finite linear combination of compact operators is a compact operator.

Prove that the composition of a compact and a bounded operator (in either order) is a compact operator.

Prove that if $\left\{ A_{n}\right\}_{1}^{\infty }$ are compact operators and $A_{n} \rightrightarrows A$, then $A$ is also a compact operator.

(J. Calkin, 1941). Let H be a separable Hilbert space. Prove that the unique nontrivial proper closed two-sided ideal in $\mathcal{B}(H)$ is $\mathcal{K}(H)$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y)$. Prove that $A$ can have a right inverse operator only in the case $\operatorname {dim} Y < \infty$. Prove that $A$ can have a left inverse operator only in the case $\operatorname {dim} X < \infty$.

Give an example of a sequence of compact operators $\left\{ A_{n}\right\}_{1}^{\infty }$ in a Banach space $X$ such that $A_{n} \xrightarrow {s} A \in \mathcal{B}(X)$, but $A$ is not a compact operator.

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$. Prove that if $\operatorname {dim}(\operatorname {Im} A) < \infty$, then $A$ is compact.

Let $X$ be a Banach space, $A \in \mathcal{K}(X, Y)$, and let $Y_{0} \subset \operatorname {Im} A$ be a closed subspace. Prove that $\operatorname {dim} Y_{0} < \infty$.

Give an example of Banach spaces $X$ and $Y$ and a non-compact operator $A \in \mathcal{B}(X, Y)$ such that $\operatorname {Im} A$ contains no infinite-dimensional closed subspaces.

Let $A \in \mathcal{B}(X)$, where $X$ is a Banach space, and $\operatorname {Im} A = X$. Can the operator $A$ be compact?

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y), B \in \mathcal{B}(X, Y)$ and $\operatorname {Im} B \subset \operatorname {Im} A$. Prove that $B \in \mathcal{K}(X, Y)$.

Let $X$ and $Y$ be normed spaces, the operator $A \in \mathcal{K}(X, Y)$. Prove that the space $(\operatorname {Im} A,\left\| \cdot \right\|_{Y})$ is separable.

Let $X$ be a Banach space. Prove that the identity operator on it is compact if and only if $X$ is finite-dimensional.

Prove that a projection operator in a Banach space is compact if and only if it is finite-dimensional.

Let $X$ be a reflexive Banach space, let $X^{*}$ be separable, and $A \in \mathcal{K}(X)$. Prove that the image of the closed unit ball $A(\bar{B}(0,1))$ is a compact set. ¹

Theorem 14.4 (S. Banach, 1932; L. Alaoglu, 1940): Let $X$ be a linear normed space. The closed unit ball of $X$ is compact in the weak topology if and only if $X$ is reflexive.

Let $X$ be a Banach space. Prove that if for every operator $A \in \mathcal{K}(X)$ the image of the closed unit ball $A(\bar{B}(0,1))$ is a compact set, then the space $X$ is reflexive.

Is it true that if in an infinite-dimensional normed space $A^{2} = 0$, then the operator $A$ is compact?

Let $X$ be an infinite-dimensional Banach space, the operator $A \in \mathcal{K}(X)$, and the polynomial $P(z) = a_{n} z^{n}+a_{n-1} z^{n-1}+\ldots +a_{0}$. Prove that if $a_{0} \neq 0$, then $P(A) \neq 0$. For an arbitrary polynomial $P(z)$ with coefficient $a_{0} = 0$, give an example of a compact operator $A$ for which $P(A) = 0$.

Prove that similar operators (see Definition 9.4) are either both compact or both non-compact.

Definition 9.4: Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X)$, $B \in \mathcal{B}(Y)$. The operators $A$ and $B$ are called similar if there exists a bounded operator $S$ with bounded inverse (that is, $S \in \mathcal{I B}(Y, X)$) such that $A S = S B$ (notation $A \sim B$).

Prove that a compact operator carries every weakly convergent sequence into a sequence convergent in norm.

Prove that an operator in a reflexive space carrying every weakly convergent sequence into a sequence convergent in norm is compact. Give an example of an operator in a nonreflexive space carrying every weakly convergent sequence into a sequence convergent in norm, but which is not compact.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y)$, the sequence $x_{n} \rightharpoonup x$ in $X$, the sequence $f_{n} \stackrel{*}{\rightharpoonup } f$ in $Y^{*}$. Prove that $f_{n}\left(A x_{n}\right) \rightarrow f(A x)$.

Let $X$ and $Y$ be Banach spaces, with $Y$ separable. Let $A \in \mathcal{B}(X, Y)$, and let the operator $A^{\prime }$ carry every $*$-weakly convergent sequence in $Y^{*}$ into a sequence convergent in norm in $X^{*}$. Prove that $A$ is compact.

Let a Banach space $X$ be reflexive, and let $X^{*}$ be separable. Prove that every compact operator $A \in \mathcal{K}(X)$ attains its norm on the closed unit ball ¹. Give an example of a compact operator in a nonreflexive space that does not attain its norm on the closed unit ball.

Theorem 14.4 (S. Banach, 1932; L. Alaoglu, 1940): Let $X$ be a linear normed space. The closed unit ball of $X$ is compact in the weak topology if and only if $X$ is reflexive.

Prove that $\operatorname {Sp}(X, Y)$ is a (in general, non-closed) linear subspace of $\mathcal{K}(X, Y)$. Prove that $\operatorname {dim} \operatorname {Sp}(X, Y) = \infty$, if $\operatorname {dim} X = \infty$ or $\operatorname {dim} Y = \infty$.

On the linear space $\operatorname {Sp}(X, Y)$ introduce the norm

where the infimum is taken over all representations of the operator $A$ as the sum of an absolutely convergent series $A = \sum_{k = 1}^{\infty } A_{k}$ of one-dimensional operators. Prove that this function satisfies the axioms of a norm. Prove the estimate $\left\| A\right\|_{\mathcal{B}(X, Y)} \leqslant \left\| A\right\|_{\mathrm{Sp}(X, Y)}$. Prove that the space $\operatorname {Sp}(X, Y)$ is Banach with respect to the introduced norm.

Prove that any Banach space in which there is a countable Schauder basis has the approximation property. Prove that any (not necessarily separable) Hilbert space has the approximation property.

Suppose that a Schauder basis exists in a Banach space $X$. Prove that the closure of the subspace of finite-dimensional operators in the sense of strong operator convergence in $\mathcal{B}(X)$ coincides with $\mathcal{B}(X)$.