The Distance from a Rank Projection to the Nilpotent Operators on

Building on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ , we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$ . For each $n\geq 2$ , we construct examples of pairs $(Q,T)$ where Q is a projection of rank $n-1$ and $T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.

On n-quasi-$[m,C]$-isometric operators

For positive integers m and n, an operator $T \in B ( H )$T∈B(H) is said to be an n-quasi-$[m,C]$[m,C]-isometric operator if there exists some conjugation C such that T∗n(∑j=0m(−1)j(mj)CTm−jC.Tm−j)Tn=0. In this paper, some basic structural properties of n-quasi-$[m,C]$[m,C]-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi-$[m,C]$[m,C]-isometric operator is again an n-quasi-$[m,C]$[m,C]-isometric operator. Moreover, we show that the class of n-quasi-$[m,C]$[m,C]-isometric operators is norm closed. Finally, we examine the stability of n-quasi-$[m,C]$[m,C]-isometric operator under perturbation by nilpotent operators commuting with T.

Some properties of (m,C)-isometric operators

In this paper, we study if T is an (m,C)-isometric operator and CT+C commutes with T, then T+ is an (m,C)-isometric operator. We also give local spectral properties and spectral relations of (m;C)-isometric operators, such as property (?), decomposability, the single-valued extension property and Dunford?s boundedness. We also investigate perturbation of (m,C)-isometric operators by nilpotent operators and by algebraic operators and give some properties.

Quasi-nilpotent perturbations of the generalized Kato spectrum

In this paper, we show that the generalized Kato spectrum of a bounded operator in a Banach space is invariant under perturbation by commuting quasi-nilpotent operators, and the Kato spectrum is stable under additive commuting nilpotent perturbations. Our results are used to give an equivalent definition of the generalized Kato spectrum.