The existence of invariant subspaces for operators with nonsymmetric growth of the resolvent

One proves the existence of invariant and hyperinvariant subspaces for certain new classes of continuous operators in a Banach space. These classes are defined by conditions on the spectrum- it has to be "thin" (while, in the interesting cases, a one-point set) - and by estimates of the resolvent (necessarily nonsymmetric). For example, one can take operators T such that σ(T)={0} and for some β ε (0,π),[Figure not available: see fulltext.] The hyperinvariant subspaces have the form Ker f(T), and f(T) is defined in some special operator calculus, constructed in the paper.