It is well-known that hyponormal operators have many interesting properties, for example, if the restriction $$T|_{{\mathcal {M}}}$$ of the hyponormal operator T on its nontrivial closed invariant subspace $${\mathcal {M}}$$ is normal, then $${\mathcal {M}}$$ reduces T. In order to discuss the reducibility of invariant subspaces of an operator, four properties of invariant subspaces ( $$R_{i}, i=1,\ldots ,4$$ ) are introduced. Among others, it is proved that, for a k-quasi- A( n) operator T, if the restriction $$T|_{{\mathcal {M}}}$$ is normal and injective, then $${\mathcal {M}}$$ reduces T, thus the function $$\sigma :T\longmapsto \sigma (T)$$ is continuous on the class of k-quasiclass- A( n) operators. Some examples related to class A( n) and n-paranormal operator are given which imply that the inclusion relations are strict. [ABSTRACT FROM AUTHOR]