If $H$ is an $\infty$-dimensional Hilbert space and $T:H\to{H}$ is a compact operator with closed range, how do I show that $T$ has finite rank, without using the open-mapping theorem? (The open-mapping theorem is not in my lecture notes).
The definitions I have in my lecture notes are:
(Let $B(H)$ denote the space of all bounded operators mapping $H\to{H}$, $K(H)$ denote the space of all compact operators mapping $H\to{H}$, $R(H)$ denote the space of all finite rank operators mapping $H\to{H}$).
- $T\in{B(H)}$ is compact if the closure of $T(B(0,1))$ is a compact set.
- $T\in{B(H)}$ has finite rank if $Range(T)=T(H)$ is finite-dimensional.
I'm not sure how to do the proof, but I think that the following propositions in my lecture notes could be useful:
- $T\in{R(H)}$ iff $T\in{B(H)}$ is the norm limit of a sequence of finite rank operators, i.e. $K(H)$ is the closure of $R(H)$.
- Let $T\in{R(H)}$. Then there is an orthonormal set $\{e_1,…,e_L\}$ s.t.
$$Tu=\sum\limits_{i,j=1}^{L}{c_{ij}(u,e_j)e_i}$$
where $c_{ij}$ are complex numbers.
Thank you in advance.
Best Answer
Sketch: Find a way to write the range of $T$ as a countable union of compact sets $K_i$. Then the Baire category theorem will guarantee that one of the $K_i$ has nonempty interior (relative to the range of $T$). This means the range of $T$ is locally compact, hence finite dimensional.