Note on a differencing and lag operators in time series modeling

Differencing operator \Delta is defined by
\Delta X_t := X_t - X_{t-1}
Lag operator B is defined by:
B X_t := X_{t-1}
It is common to see in literature the following:
\Delta X_t = X_t - X_{t-1} = X_t - B X_t = (1 - B) X_t
Authors often omit to say that (1 - B) in the last equality is an operator whose definition is the preceding equation; the last equality in above should better read as:
X_t - B X_t := (1 - B) X_t
Otherwise, one can ask, how the hell subtraction of operator from a real number is supposed to be carried out ?! 🙂
If we want to be more formal in definition of operators, which are maps, we should specify domains and ranges of maps:
\Delta : \mathcal{R}^w \mapsto \mathcal{R}^w
B : \mathcal{R}^w \mapsto \mathcal{R}^w
This is a formal way to say, that the domain and range is a set of all sequences of real numbers.

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