range of linear operator on infinite dimensional when null space is 0

Question

Consider a linear operator $A: H\rightarrow H$.

When $H$ is finite dimensional and $N_A=\{0\}$, we have $R_A = H$.

Does it still hold for an infinite dimensional case?

Since $dim N_A = dim R_A^\bot$, we can say that $dim R_A = \infty$, but this does not imply $R_A=H$.

Answer 1

Let $H=\ell^{2}$ and $T(x_n)=(0,x_1,x_2,..)$. The kernel of $T$ is $0$ but $T$ is not onto.

Another example is $f \to \int_0^{x}f(t)dt$ on $C[0,1]$.