Iterated Aluthge transforms: a brief survey. (English) Zbl 1151.37029
Summary: Given an \(r\times r\) complex matrix \(T\), if \(T=U|T|\) is the polar decomposition of \(T\), then the Aluthge transform is defined by
\[ \Delta(T)= |T|^{1/2} U|T|^{12}. \]
Let \(\Delta(T)\) denote the \(n\)-times iterated Aluthge transform of \(T\), i.e. \(\Delta^0(T)=T\) and \(\Delta^n(T)= \Delta(\Delta^{n-1}(T))\), \(n\in\mathbb N\). In this paper we make a brief survey on the known properties and applications of the Aluthge trasnsorm, particularly the recent proof of the fact that the sequence \(\{\Delta^n(T)\}_{n\in\mathbb N}\) converges for every \(r\times r\) matrix \(T\). This result was conjectured by I. B. Jung, E. Ko and C. Pearcy [Integral Equations Oper. Theory 45, No. 4, 375–387 (2003; Zbl 1030.47021)].
\[ \Delta(T)= |T|^{1/2} U|T|^{12}. \]
Let \(\Delta(T)\) denote the \(n\)-times iterated Aluthge transform of \(T\), i.e. \(\Delta^0(T)=T\) and \(\Delta^n(T)= \Delta(\Delta^{n-1}(T))\), \(n\in\mathbb N\). In this paper we make a brief survey on the known properties and applications of the Aluthge trasnsorm, particularly the recent proof of the fact that the sequence \(\{\Delta^n(T)\}_{n\in\mathbb N}\) converges for every \(r\times r\) matrix \(T\). This result was conjectured by I. B. Jung, E. Ko and C. Pearcy [Integral Equations Oper. Theory 45, No. 4, 375–387 (2003; Zbl 1030.47021)].
MSC:
| 37D10 | Invariant manifold theory for dynamical systems |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
