1
vote
1answer
37 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
3
votes
2answers
48 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
3
votes
1answer
65 views

Show compactness/ noncompactness of an operator by approximation

I have to show whether the following operator is compact or not: $$ T\colon\ell^2\to\ell^2: (x_n)_{n\in\mathbb{N}}\mapsto\left(\frac{x_n+x_{n+1}}{2}\right) $$ My idea was to approximate $T$ by ...
1
vote
2answers
56 views

Determine operator norm and show compactness

Consider $$ T\colon\ell^1\to\ell^1, (s_n)\mapsto\left(\frac{s_{n+1}}{n}\right). $$ Calculate the norm of $T$ and show that $T$ is compact. 1.) Operator norm of $T$ What I have is the ...
0
votes
1answer
43 views

Show non-compactness of multiplication operator on $C[0,1]$

Show that the multiplication operator $$ (Ax)(t):=(t+1)x(t) $$ in the Banachspace $C[0,1]$ is not compact. Again I am struggling with compactness, it is always difficult to me to decide ...
4
votes
1answer
89 views

Continuous, selfadjoint and compact?

Hell0 there! I have to show whether the operator $$ T\colon L^2(\mathbb{R})\to L^2(\mathbb{R}), f\mapsto\chi_{[0,1]}f $$ is continuous, selfadjoint and compact. I have problems to show the ...
2
votes
0answers
61 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
8
votes
2answers
229 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...