Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

Spectral properties of square hyponormal operators

In this paper, we introduce a square hyponormal operator as a bounded linear operator T on a complex Hilbert space H such that T2 is a hyponormal operator, and we investigate some basic properties of this operator. Under the hypothesis ?(T) ? (-?(T)) ? {0}, we study spectral properties of a square hyponormal operator. In particular, we show that if z and w are distinct eigen-values of T and x,y ? H are corresponding eigen-vectors, respectively, then ?x,y? = 0. Also, we define nth hyponormal operators and present some properties of this kind of operators.

On Subscalarity of Some 2 × 2M-Hyponormal Operator Matrices

We provide some conditions for2×2operator matrices whose diagonal entries areM-hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.