Give an example of a linear operator T on an inner product space V such that $N(T) \neq N(T^{*})$

Question

Problem: Give an example of a linear operator T on an inner product space V such that $N(T) \neq N(T^{*})$

Any general idea or example for this? How am I supposed to think about this?

Answer 1

There are some basic things that can be noted, but after this you just have to try some examples. Firstly, lets take user744868's comment, and consider real square matrices, and see if we can find one whose transpose has a different nullspace.

Obviously we don't want to take an invertible matrix, as its nullspace will be empty (and therefore the nullspace of its transpose will also be empty). So lets try one which will clearly have a nontrivial nullspace, lets say $$A = \left[\begin{array}{cc} 1 & 1 \\ 2 & 2 \end{array} \right].$$ Setting $Ax = 0$, and solving using row echelon operations (or otherwise), we see that $x_1 = -x_2$, and so a basis for the nullspace is $\left\{ \left(\begin{array}{c} 1 \\ -1 \end{array} \right)\right\}$. Now consider the transpose of $A$, and see if its nullspace is different!

Answer 2

Consider $H=\ell^{2}$. Let $T(a_n)=(a_2,a_3,...)$. Then $T^{*}(a_n)=(0,a_1,a_2,...)$. So $N(T)$ is the one-diemensional space spanned by $(1,0,0,...)$ and $N(T^{*})=\{0\}$.

PS: Null spaces of $T$ and $T^{*}$ are 'rarely' equal.