# A compact, bounded, closed-range operator on a Hilbert space has finite rank

1. May 26, 2012

### SiennaB

1. The problem statement, all variables and given/known data

Let $H$ be an $\infty$-dimensional Hilbert space and $T\to{H}$ be an operator.

Show that if $T$ is compact, bounded and has closed range, then $T$ has finite rank. Do not use the open-mapping theorem.

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 $H\to{H}$.

2. Relevant defintions

*$T\in{B(H)}$ 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.

3. The attempt at a solution

I'm not sure how to do the proof, but I think 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 ${{e_1,...,e_L}}\in{H}$ s.t. $$Tu=\sum\limits_{i,j=1}^{L}{c_{ij}(u,e_j)e_i}$$ where $c_{ij}$ are complex numbers.

Any help with the proof would be greatly appreciated.

Thank you in advance.

Last edited: May 26, 2012
2. May 26, 2012

### micromass

Staff Emeritus
Do you know that compact operators can never be invertible on Hilbert spaces??

Given your compact operator T with closed range, can you modify the domain and codomain a bit such that T becomes a bijection between Hilbert spaces??

3. May 26, 2012