# Example of Closed Linear Operator whose range is not closed

I had found example of Linear operator whose range is not closed.

But I am intersted in finding exmple of closed operator (which has closed graph) but do not have closed range.

Please can anyone give me hint to find such example

Thanks a lot

Let $$Lf = \int_0^x f(t)dt$$ be defined on $$C[0,1]$$ with the sup norm. $$L : C[0,1]\rightarrow C[0,1]$$ is bounded and, hence, has a closed graph. $$L$$ in injective because $$Lf=0$$ for some $$f\in C[0,1]$$ implies that $$f=\frac{d}{dx}Lf=0$$.

If $$\mathcal{R}(L)$$ were closed in $$C[0,1]$$, then the closed graph theorem would imply that $$L$$ would have a bounded inverse $$L^{-1} : \mathcal{R}(L)\subset C[0,1]\rightarrow C[0,1].$$ Of course the inverse must be $$L^{-1}=\frac{d}{dx}$$. It is easy to verify that $$\frac{d}{dx}$$ is not bounded on the bounded set $$\{ x,x^2,x^3,\cdots\}\subset \mathcal{R}(L)$$. So $$\mathcal{R}(L)$$ cannot be closed.

If $$(a_n)\subset\mathbb C$$ with $$a_n\neq 0$$ for all $$n\in\mathbb N$$ is unbounded such that also $$(a_n^{-1})$$ is unbounded, then the operator $$T$$ in $$\ell^2$$, defined by $$(Tx)_n = a_nx_n$$ with $$\text{dom}T = \{x\in\ell^2 : (a_nx_n)\in\ell^2\}$$ is closed and has a non-closed range. This is pretty easy to see: the operator is a multiplication operator, so it's closed and densely defined. Its inverse is given by $$(T^{-1}x)_n = a_n^{-1}x_n$$ with the appropriate domain. Its domain is dense (also a multiplication operator) and coincides with the range of $$T$$.

• Indeed, if you take the $a_n$ to be bounded instead, with $a_n^{-1}$ still unbounded (for instance $a_n = 1/n$), your operator $T$ is bounded and everywhere defined (hence definitely closed), and still its range is not closed. – Nate Eldredge Sep 6 '19 at 15:52
• @NateEldredge Sure, but I assumed that OP would like to see an unbounded operator with this property. – amsmath Sep 6 '19 at 15:55

Any compact operator of infinite rank is an example.

Some examples:

• the inclusions $$\ell^p\to \ell^q$$, $$L^q[0,1]\to L^p[0,1]$$ with $$p
• integral operators on $$C[0,1]$$ of the form $$f\mapsto (x\mapsto \int_0^x f(x)g(x)\,\mathrm{d}x)$$ for some nonzero $$g\in C[0,1]$$,
• any map $$\ell^p\to \ell^p$$ defined by $$x\mapsto xy$$, where $$y$$ converges to $$0$$ (and has infinite support).