I would like to construct a compact linear operator $A:X\to Y$ such that $$ (I-A)\varphi = 0 $$ for all $\varphi = a_0 + a_1x$, i.e. for all polynomials of degree 1? Here $X$ and $Y$ should be some function spaces on the domain $[0,1]$. This is because I want an intuitive example of an operator $L=I-A$ where $A$ is compact and $L$ has a two-dimensional nullspace.
Is it possible to specify an operator $A$ and spaces $X$ and $Y$ such that the above holds?
Take $X = Y = C^0[0,1]$ (with the sup-norm), and look at the linear operator $L:X\to Y$ given by $$ (Lf)(x) = f(x) - f(1)x + (x-1)f(0) $$ which, geometrically, shears and vertically translates the graph of $f$ so that $(Lf)(0) = (Lf)(1) = 0$. This makes $$ (Af)(x) = f(1)x - (x-1)f(0) $$ so $A$ gives the straight line between $(0,f(0))$ and $(1, f(1))$.