Triangular n-isometric operators

Abstract

In this paper, we introduce the class of triangular n-isometric operators and study various properties. We show that every triangular n-isometric operator is subscalar of order 2n; in particular, every isometric operator is subscalar of order two. Consequently, if the spectrum of a triangular n-isometric operator T has a nonempty interior in $\mathbb{C}$, then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for triangular n-isometric operators. Some spectral properties of this class of operators are also presented.

Keywords:

• Isometric operator
• subscalar operator
• invariant subspace
• hyperinvariant subspace
• Weyl’s theorem

AMS Subject Classifications:

• Primary: 47A11
• Secondary: 47A15
• 47B20

Disclosure statement

No potential conflict of interest was reported by the authors.