Existence of invariant subspaces for operators with non-symmetrical growth of resolvent

B. M. Solomyak

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 204-208

Résumé

Existence of invariant and hyperinvariant subspaces is obtained for some new classes of bounded operators in a Banach space. The operators under consideration have “thin” spectrum (in the most interesting cases the spectrum is a single point) and a certain nonsymmetry in the growth of resolvent. For example, one can take $T$ such that $\sigma(T)=\{0\}$ and for some $\beta\in(0,\pi]$, \begin{gather} \|(\lambda J-T)^{-1}\|\le c|\lambda|^{-n},\quad|\arg\lambda|>\beta;\\ \quad\|(\lambda J-T)^{-1}\|\le c\exp|\lambda|^{-\pi/2\beta}, \quad|\arg\lambda|\le\beta. \end{gather} Hyperinvariant subspaces have the form $\operatorname{Ker}f(T)$, where $f(T)$ is defined in a special functional calculus.