Cancellation operation in time series analysis When studying the time series analysis, I read the following example.

I do not know how to understand this cancellation process. Yes, it can be cancelled like normal algebraic formula. But this “z” is an operator instead of just a variable. How can we cancel the common factor like we do in algebra?
 A: Actually, $B$ is the operator in question. Or rather a linear function of it, the operator $(1+0.5B)$.
If the operator has an inverse (it does), then the cancellation will be meaningful, since it's just multiplying both sides by the inverse operator. 
That is, multiply $\phi(z)$ and $\theta(z)$ by $(1+0.5z)^{-1}$, which represents that inverse operator, which is actually meaningful (as long as you have convergence*). Cancellation complete.
* (you might consider where $(1+az)^{-1} = 1 - az + (az)^2 - (az)^3 + ...$ converges, say, and perhaps consider what might happen at a zero of a term like $(1+az)$)
If you write out what is going on in terms of the original series, it might be clearer.
A: It is possible to cancel the common factor as in algebra because it is possible to prove it that it works. 
The idea is to to start with the series $\sum_j \psi_j X_{t-j}$, where $X_{i}$ is the time series and show that this series converges if $\sum_j |\psi_j|<\infty$ and $\sup_tE|X_t|<\infty$. Now the convergence of series $\sum_j |\psi_j|<\infty$ implies that the Laurent series $\sum_j \psi_j z^j$ converges absolutely on unit circle $\{z\in \mathbb{C}: |z|=1\}$ and hence defines function $\psi(z)$. 
So you have relation between the the polynomials and time series and hence the rules of algebra are "transferable". So if you have $\phi(B)X_t=\theta(B)Z_t$ you can write $X_t=\psi(B)Z_t$, where $\psi(z)=\frac{\theta(z)}{\phi(z)}$. If $\theta(z)$ and $\phi(z)$ have common roots, they cancel themselves in the fraction as in usual algebra. Note however that the cancelation is possible when the remainder (after cancelation) $\frac{\theta(z)}{\phi(z)}$ is well defined, i.e. $\phi(z)$ must not have roots in the unit circle, as only then the equation $\phi(B)X_t=\theta(B)Z_t$ has a well defined solution.
If you want more details I would recommend reading these excellent lecture notes by A. Van der Vaart. My answer can be thought as a condensed summary of the first pages of chapter on ARIMA processes.
