Articles of compact operators

Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition – that is, either that $V$ maps bounded sets to precompact sets, or equivalently, that for any bounded sequence $(f_n)$ in the domain, $\{Vf_n\}$ has a convergent subsequence. I have seen an […]

Rellich's theorem for Sobolev space on the torus

From John Roe: Elliptic operators, topology and asymptotic methods, page 73: Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, f_{2}\rangle_{k}=(2\pi)^{k}\sum_{v\in \mathbb{Z}^{n}}\tilde{f}_{1}(v)\overline{\tilde{f}_{2}}(v)(1+|v|^{2})^{k} $$ John Roe claimed that there is a Rellich type compact embedding theorem available. If $k_{1}<k_{2}$, then the inclusion operator $H^{k_{2}}\rightarrow H^{k_{1}}$ is a […]

Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle T(x),y\rangle$). Now someone has proved the below theorem. Is it true? BEGIN Let $X$ and $Y$ be two arbitrary Banach spaces. Then $F (X; Y^*)$(Finite rank operator) is dense […]

True/False: Self-adjoint compact operator

Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is true that $T$ has to be injective on the whole Hilbert space? Until now, I […]

Compact operator on $l^2$

Let A be a bounded linear operator on $l^2$ defined by A($a_n$)=($\frac{1}{n} a_n$). Would you help me to prove that A is compact operator. I guess the answer using an approximation by a sequences of finite range operator.

Compact operator with closed range has finite dimensional range

Let $X,Y$ be Banach Spaces, and let $T\in K(X,Y)$ be a compact operator from $X$ to $Y$. I have to prove that $T(X)$ is closed in Y if, and only if, $\dim(T(X))<\infty$. Can anybody help me with this proof, please? There is surely some property I haven’t thought about, but I’m getting really weird right […]

finite dimensional range implies compact operator

Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in X$ is a bounded sequence, there exist a subsequence $x_{n_k}$ such that $Tx_{n_k}$ is convergent. I want to prove that if $T\colon […]

Compact operators and completely continuous operators

A compact operator between Banach spaces is an operator that maps bounded sets into relatively compact sets, while a completely continuous operator maps all weakly convergent sequences into convergent sequences. Compact operators are always completely continuous, but completely continuous operators may be non-compact: the identity operator in the Schur space ${\rm l}_1$ is an example. […]

Inequalities on kernels of compact operators

Suppose we have a $\sigma$-finite positive measure $\mu(v)$ on $\Bbb R^d$ and we have two positive kernels on $\Bbb R^d\times \Bbb R^d$ $k_1(v,u)>0$, $k_2(v,u)>0$. We define integral operators $$K_i[f](v) =\int_{\Bbb R^d}k_i(v,u)f(u)d\mu(u).$$ We know by some external argument that $K_2:L^2(d\mu)\to L^2(d\mu)$ is a compact operator; furthermore, we know that there exist two strictly positive constants $C_+$ […]

Rellich–Kondrachov theorem for traces

Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let furthermore $\gamma$ be the trace map. I am looking for a theorem that allows me to conclude that $\gamma \colon W^{1,p}(\Omega) \to L^q(\partial \Omega)$ is compact whenever $q […]