I learned that there is a holomorphic functional calculus for closed operators: If $T$ is a closed operator on a Hilbert space, and $f$ is a function that is holomorphic on some open subset $\Omega$ of $\mathbb{C} \cup \{\infty\}$ that contains $\infty$ if $T$ is unbounded, then I can form $f(T)$.

Does there also exist a continuous or measurable functional calculus, for a non-self-adjoint closed operator?

If not, what are the problems?

P.S.: The problem is that if I understand the results correctly, $\Omega$ has to contain $\infty$ if $T$ is unbounded, so I can never form $f(T)$ for an entire function, unless $f$ is a polynomial entire function. I would, however, very much like to take $\cos(T)$, for example :->

normalclosed operators there is. $\endgroup$Perturbation theory of linear operatorsor the classic 3 volume by Dunford and SchwartzLinear Operators$\endgroup$