range and kernel of linear transformation over infinite dimensional vector spaces

Question

How to find the range and kernel of such linear transformations ? I have already gone over the literature and have found some useful helps at example 1 and example 2. However they deal with finite dimensional vector spaces and I am working on infinite dimensional vector spaces.

1. $T:P \rightarrow P$ defined by $T(p)(x) = x p(x)$
2. $T:P \rightarrow P$ defined by $T(p)(x) = x p'(x)$
3. $T:P \rightarrow P$ defined by $T(p)(x) = p''(x) - 2p(x)$

I have tried to solve the problems and I am stating below what I tried so far :

1. Let $p \in P$, then $p = a_0 + a_1x + a_2 x^2 + a_3 x^3 + ....$.

Then $T(p)(x) = xp(x) = x(a_0 + a_1 x + a_2 x^2 + ....)=(a_0 x + a_1 x^2 + a_2 x^3 + ....)$. So basically it means that a polynomial vector in this representation $(a_0,a_1,a_2,a_3,...)$ is getting transformed into a new polynomial vector as $(0,a_0,a_1,a_2,a_3,...)$. I understood upto this point but could not figure how to compute the range of $T$ i.e., $R(T)$.

Regarding nullity, I computed $T(p)(x)=0 => (a_0x + a_1x^2 + a_2 x^3 + ....) = (0,0,0,.....)$ which tells me that $a_0=a_1=a_2=a_3=....=0$. So Null space of $T$ is $N(T)={0}$ and $dim(N(T)) = 0$.

Please help me to figure out to compute the $R(T)$.

1. For the second problem I followed exactly the same steps as the first one. Let $p \in P$. Then $T(p)(x) = xp(x) = x(a_0 + a_1x + a_2x^2 + ...)' = x(a_1 + 2a_2x + 3a_3x^2 + ....) = (a_1x + 2a_2x^2 + 3a_3x^3 + ....)$.

So a polynomial vector of this form $(a_0,a_1,a_2,a_3,...)$ is getting transformed into $(0,a_1,2a_2,3a_3,....)$. Again I could not figure out $R(T)$.

Regarding nullity I could easily check that : $T(p) = 0 $ happens if $a_0 = $constant and $a_1 = a_2 = a_3 = ... = 0$. So $N(T) =$ constant polynomial.

3. I could not figure out either how to compute either $R(T)$ or $N(T)$.

Please guide me how to solve this problem. It will be or more help to me if you can tell me specifically how to solve such problems for such infinite dimensional vector spaces like $C(0,1)$ i.e, the set of all continuous real valued functions defined over the open interval $(0,1)$. Also please point me out if my approaches are wrong.

Answer 1

I sahll assume that you are considering polynomials with coefficients in $\mathbb R$ (or $\mathbb Q$ or $\mathbb C$ or similar)

1. You have noticed that all elements of $R(T)$ have zero constant term. In fact $R(T)$ is the space of polynomials with zero constant term (or equivalently polynomials $q$ with $q(0)=0$): From any such polynomial $q$ we can factor out $x$, i.e., write it as $q(x)=xp(x)$ and thus obtain $p$ that maps to $q$.

2. Here the null space is larger than in the previous part for $xp'$ for all constant $p$ (and only for those, why?). The range however is the same, i.e., the space of polynomials with zero constant term: As above, we can factor $x$ from any such polynomial, and we can (formally) integrate the remaining factor. The polynomial obtained this way is (clearly) mapped to $q$.

3. This $T$ is injective because we have $\deg p''<\deg p$, hence $\deg (p''+2p)=\deg p$. $T$ is also surjective: Given any polynomial $q$ we can find $p$ with $T(p)=q$ by checking the coefficients from top to bottom; in fact if $q(x)=\sum a_kx^k$ and we make the ansatz $p(x)=\sum_k b_kx^k$ then we obtain equations $(k+2)(k+1)b_{k+2}-2b_k=a_k$, i.e., $b_k=\frac{(k+1)(k+2)b_{k+2}-a_k}{2}$ where we can consider the right hand side known.