Inversion of linear combination of discrete shift operators

Question

I have recently tackled the following problem and I'm seeking for some help.

Let me define the shift operator \begin{equation} T_h[\cdot], h \in \mathbb{Z} \end{equation} such that \begin{equation} T_h[f(a)] = f(a+h) \end{equation}

where $a \in \mathbb{Z}$(including $0$) and $f: \mathbb{Z} \mapsto \mathbb{R}$.

My problem is the following: does the inverse ($\mathbb{L}^{-1}$) of the following operator exist? If so is there a method to calculate it?

\begin{equation} \mathbb{L}[\cdot] = (pT_1 + k T_{-1})[\cdot] \end{equation}

where $p,k \in \mathbb{R}$.

Thank you all in advance for possible answers.

Answer 1

If $f$ has nice enough properties, a $z$ transform should work. Namely, consider $$G(z)=\sum_{n=-\infty}^{\infty} z^n f(n)$$

Answer 2

Notice that, since $T_1$ is invertible, your operator is invertible iff $pT_1^2+k$ is invertible, iff $T_1^2+k/p$ is invertible, iff $-k/p$ lies in the resolvent of $T_1^2$.

Now you need to specify precisely which space is taken for the domain of $T$, but in case you choose $\ell^2$, the spectrum of $T_1$ is the unit circle, hence the answer to your question is that your operator is invertible iff $$|p|\neq|k|.$$