Heredity for triangular operators

Abstract

A proof is given that if the lower triangular infinite matrix $T$ acts boundedly on $\ell^2$ and U is the unilateral shift, the sequence $(U^*)^nTU^n$ inherits from $T$ the following properties: posinormality, dominance, $M$-hyponormality, hyponormality, normality, compactness, and noncompactness. Also, it is demonstrated that the upper triangular matrix $T^*$ is dominant if and only if $T$ is a diagonal matrix.

Keywords

• posinormal operator
• dominant operator
• compact operator
• $M$-hyponormal operator
• hyponormal operator
• triangular matrix
• terraced matrix