# to show unboundedness of an operator

Given $$l$$ is an even, $$2\pi-$$periodic, $$L^2$$ function. An operator $$L: H \to H$$, where $$H$$ is a Hilbert space, is defined as $$Lf:=\int_{-\pi}^\pi l(x-y)f(y)dy.$$ To show that $$L$$ does not have bounded inverse.

I don't know where to begin with to solve this problem. Any steps of hints towards showing this? Thanks in advance!

Update: Does it suffice to say that, $$L$$ is compact, so $$0\in\sigma(L)$$, and thus the inverse of $$L$$ isn't bounded?

• Your argument is fine. Compact operators on an infinite dimensional HS cannot have abounded inverse. – Kabo Murphy Nov 17 at 0:22