Let $y_1,y_2$ be independent Bernoulli variables, and let $f(y_i)=ay_i+1$ be a function defined on both Bernoulli variables. Also, let $B_i(f(y_i))=ay_i$ be an operator acting on the function $f$. Note that $B_i(f(y_j))=0$ if $i\neq j$.
I want to understand the action of the operator $B_1+B_2$ on $f(\frac{y_1+y_2}{\sqrt{2}})$. Note that $f(\frac{y_1+y_2}{\sqrt{2}})=a(\frac{y_1+y_2}{\sqrt{2}})+1$
Is $B_i$ linear? If it is, we can write $a(\frac{y_1+y_2}{\sqrt{2}})+1$ as $[a\frac{y_1}{\sqrt{2}}+\frac{1}{\sqrt{2}}]+[a\frac{y_2}{\sqrt{2}}+\frac{1}{\sqrt{2}}]+1-\sqrt{2}$. Clearly, $$B_1+B_2:[a\frac{y_1}{\sqrt{2}}+\frac{1}{\sqrt{2}}]+[a\frac{y_2}{\sqrt{2}}+\frac{1}{\sqrt{2}}]\to a\left(\frac{y_1}{\sqrt{2}}+\frac{y_2}{\sqrt{2}}\right)$$ But how does $B_1+B_2$ act on the remaining $1-\sqrt{2}$?