Journal of the
Korean Mathematical Society JKMS

A Banach space operator $A\in\mathcal X$ is polaroid if the isolated points of the spectrum of $A$ are poles of the resolvent of $A$; $A$ is hereditarily polaroid, $A\in (\mathcal H\mathcal P)$, if every part of $A$ is polaroid. Let $\mathcal X^n=\bigoplus_{i=1}^n{\mathcal X_i}$, where $\mathcal X_i$ are Banach spaces, and let ${\mathcal A}$ denote the class of upper triangular operators $A=(A_{ij})_{1\leq i,j\leq n}$, $A_{ij}\in B(\mathcal X_j,\mathcal X_i)$ and $A_{ij}=0$ for $i>j$. We prove that operators $A\in{\mathcal A}$ such that $A_{ii}$ for all $1\leq i\leq n$, and $A^*$ have the single--valued extension property have spectral properties remarkably close to those of Jordan operators of order $n$ and $n$-normal operators. Operators $A\in{\mathcal A}$ such that $A_{ii}\in (\mathcal H\mathcal P)$ for all $1\leq i\leq n$ are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, $A+R$ satisfies Browder's theorem for all upper triangular operators $R$, such that $\oplus_{i=1}^n{R_{ii}}$ is a Riesz operator, which commutes with $A$.

Keywords: Banach space, $n$-normal operator, hereditarily polaroid operator, single valued extension property, Weyl's theorem

MSC numbers: Primary 47B47, 47A10, 47A11