Examples of bounded linear operators with range not closed

Question

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range?

Thanks!

Answer 1

Define $T:L^1(\Bbb R)\to L^1(\Bbb R)$ by $Tf = gf$, where $$g(t)=\frac{1}{1+t^2}.$$Functions with compact support are dense in $L^1$, hence $T(L^1)$ is dense in $L^1$. But you can easily find an example showing that $T(L^1)\ne L^1$; hence $T(L^1)$ is not closed in $L^1$.

Answer 2

This is an example from Booss Bleecker. Let $\mathcal{H}:=l^2(\mathbb{N}_{\geq1})$ and $\{\delta_x\}_{x\in\mathbb{N}_{\geq1}}$ be the orthonormal basis of $\mathcal{H}$. Define $A:\mathcal{H}\to\mathcal{H}$ by: $$ A:=\sum_{x\in\mathbb{N}_{\geq1}}\frac{1}{x}\delta_x<\delta_x,\cdot>$$

Verify that:

1. $A\in\mathcal{B}(\mathcal{H})$ (with $||A||=1$)
2. $v\in im(A)$ if and only if $\sum_{x\in\mathbb{N}_{\geq1}}x^2|<\delta_x,v>|^2 < \infty$.
3. With $v_0 := \sum_{x\in\mathbb{N}_{\geq1}}\frac{1}{x^{\frac{3}{2}}}\delta_x$ and $\{v_j\}_{j\in\mathbb{N}_{\geq1}}$ with $v_j := \sum_{x\in\mathbb{N}_{\geq1}}\frac{1}{x^{\frac{3}{2}+\frac{1}{j}}}\delta_x$ we have $v_0\notin im(A)$ yet $v_j \in im(A)$ and $v_j\to v_0$.

Answer 3

Take an operator $T\colon \ell_\infty \to c_0$ given by

$$T(\xi_n)_{n=1}^\infty = (\frac{\xi_n}{n})_{n=1}^\infty\quad (\xi_n)_{n=1}^\infty\in \ell_\infty).$$

Certainly $T$ is injective and has dense range. However, it cannot have closed range as it would be an isomorphism which is impossible as $\ell_\infty$ is non-separable.

This operator is actually compact. Compactness is a handy condition thay prevents operators whose range is not finite-dimensional to have closed range.