I want to find a (non-trivial, closed) invariant subspace for the operator $T: C[0,1] \rightarrow C[0,1]$ defined by
$(Tf)(x) = \displaystyle\int_0^x2tf(t)dt + f(1)$
The usual candidates don't seem to work though. I tried:
$A_n := \{f$ is a polynomial of degree $\le n\}$ ($exp$, $sin$,... quickly fail too),
$B_y := \{f: f|_{[0,y]}=0\}$ (and some variations, including $f(1)=0$)
As a relative of the Volterra operator, some other form of $B_y$ should probably do the trick but I can't figure it out. Integration by parts didn't help much either:
$(Tf)(x) = f(1) + \displaystyle2x\int_0^xf(t)dt-2\int_0^x\int_0^tf(s)dsdt = f(1) +2x*(Vf)(x)-2*(V^2f)(x)$