The following main theorem is proved:
Theorem 4. If T is a hyponormal operator acting on a given infinite dimensional, separable, complex Hilbert space, and $R(\sigma(T))\ne C(\sigma (T))$, then T has an invariant subspace where $C(\sigma(T))$ denotes the space of continuous functions on $\Sigma(T)$, the spectrum of T, and $R(\sigma(T))$ denotes the closure in $C(\sigma(T))$ of the rational functions with poles off $\sigma$ (T).
The invariant subspace proof for subnormal operators obtained by the author [Integral Equations Oper. Theory 1, 310-333 (1978;

Zbl 0416.47009)] does not adapt to hyponormal situation.
The result stated above is reformulated, in the equivalent result that follows: If T is a hyponormal operator and G is a non-empty open subset of the complex plane such that $\sigma(T)\cap G$ is dominating in G, then T has a nontrivial invariant subspace.
{\it C. Apostol} [Rev. Roumaine Math. Pures Appl. 13, 1481-1528 (1968;

Zbl 0176.437)] proved the equivalence of these statements for subnormal operators.
The present paper used the techniques of Apostol and makes them work for the more general operators on a Hilbert space that are decomposable but not unconditionally decomposable, and in the process the invariant subspace result for hyponormal operators is achieved.