I'm reading about Hodge Theory from Griffiths and Harris. The setup is as follows.
$M$ is a compact Kahler manifold with Kahler form $w$. We have the well known operators $d , \partial , \bar{\partial}$. We define another set of operators $L$ and $\Lambda$. $L(\theta) = \theta \wedge w$. And $\Lambda$ is the adjoint of $L$.
Then they go on to write that :
"$d , d^c = i.(\bar{\partial} - \partial), \Lambda$ are real differential operators."
In what sense are these real?
On
It just means that they are differential operators if you consider them as acting on real-valued functions. Here this just means they satisfy the product rule if you apply them to a product of two functions.
This is in contrast to the concept of complex differentiable which relates to functions that are holomorphic.
Too long for a comment, but not an answer.
As a brief explanation on why $\Lambda$ is real: the fundamental Kähler form can be defined as a real 2-form, i.e. as a smooth section of $\Lambda^2 T^*\underline{M}$, where $\underline{M}$ is the real manifold underlying $M$. Then you can embed these real 2-forms into the complex-valued 2-forms via the complexification map. The image under this map of the real $\omega$ happens to land on the $(1,1)$ term of the decomposition $$ \Lambda^2 T^*_\mathbb{C}M = \Lambda^{2,0} T^*M \oplus \Lambda^{1,1} T^*M \oplus \Lambda^{0,2} T^*M $$ and thus the Lefschetz operators $\Lambda,L$ are the complexification of the same real operators.
This is very thoroughly explained, as I said, in Huybrechts Section 1.2.