Power Partial Isometry Index and Ascent of a Finite Matrix

We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$-by-$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power partial isometry index $p(A)$ of a matrix $A$ is the supremum, possibly infinity, of nonnegative integers $j$ such that $I, A, A^2, \ldots, A^j$ are all partial isometries while the ascent $a(A)$ of $A$ is the smallest integer $k\ge 0$ for which $\ker A^k$ equals $\ker A^{k+1}$. It was known before that, for any matrix $A$, either $p(A)\le\min\{a(A), n-1\}$ or $p(A)=\infty$. In this paper, we prove more precisely that there is an $n$-by-$n$ matrix $A$ such that $p(A)=j$ and $a(A)=k$ if and only if one of the following conditions holds: (a) $j=k\le n-1$, (b) $j\le k-1$ and $j+k\le n-1$, and (c) $j\le k-2$ and $j+k=n$. This answers a question we asked in a previous paper.Comment: 11 page

Numerical Ranges of KMS Matrices

A KMS matrix is one of the form J_n(a)=[{array}{ccccc} 0 & a & a^2 &... & a^{n-1} & 0 & a & \ddots & \vdots & & \ddots & \ddots & a^2 & & & \ddots & a 0 & & & & 0{array}] for $n\ge 1$ and $a$ in $\mathbb{C}$. Among other things, we prove the following properties of its numerical range: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\neq 0$, (2) its boundary $\partial W(J_n(a))$ contains a line segment if and only if $n\ge 3$ and $|a|=1$, and (3) the intersection of the boundaries $\partial W(J_n(a))$ and $\partial W(J_n(a)[j])$ is either the singleton \{\min\sigma(\re J_n(a))\} if $n$ is odd, $j=(n+1)/2$ and $|a|>1$, or the empty set $\emptyset$ if otherwise, where, for any $n$-by-$n$ matrix $A$, $A[j]$ denotes its $j$th principal submatrix obtained by deleting its $j$th row and $j$th column ($1\le j\le n$), \re A its real part $(A+A^*)/2$, and $\sigma(A)$ its spectrum.Comment: 35 page

Numerical range of Aluthge transform of operator

AbstractFor any operator A on a Hilbert space, let A denote its Aluthge transform. In this paper, we prove that the closure of the numerical range of A is always contained in that of A. This supplements the recently proved case for dimkerA⩽dimkerA* by Yamazaki, and partially confirms a conjecture of Jung, Ko and Pearcy

The operator factorization problems

AbstractWe survey various results concerning operator factorization problems. More precisely, we consider the following setting. Let H be a complex Hilbert space, and let B(H) be the algebra of all bounded linear operators on H. For a given subset C of B(H), we are interested in the characterization of operators in B(H) which are expressible as a product of finitely many operators in C and, for each such operator, the minimal number of factors in a factorization. The classes of operators we consider include normal operators, involutions, partial isometries together with their various subclasses, and other miscellaneous classes of operators. Most of the known results are for operators on finite-dimensional spaces or finite matrices. The paper concludes with some applications, due to Hochwald, concerning the uniqueness of the adjoint operation on operators