Does there exist any $T \in \mathcal L (\mathcal H)$ such that $0 \lt \dim \text {ker}\ T \lt \infty$ but $\text {ran}\ T$ is not closed?

Question

Let $\mathcal H$ be a Hilbert space. Does there exist any $T \in \mathcal L (\mathcal H)$ such that $0 \lt \dim \text {ker}\ T \lt \infty$ but $\text {ran}\ T$ is not closed?

I am trying to find out an example of this but every time I get the range to be closed. May be I am not thinking in the right direction. Could anyone please help?

Thanks for your time.

Answer 1

Hint $:$ Let $\mathcal H_1 = \ell^2(\mathbb N)$ and $\mathcal H_2 = L^2[0,1].$ Let $T_1$ be the left shift on $\mathcal H_1$ and $T_2 = M_x$ where $M_x$ represents the multiplication by $x \mapsto x$ on $\mathcal H_2.$ Now consider the operator $T = T_1 \oplus T_2$ on $\mathcal H = \mathcal H_1 \oplus \mathcal H_2.$