Abstract
Associated with every operatorT on Hilbert space is its Aluthge transform\(\tilde T\) (defined below). In this note we study various connections betweenT and\(\tilde T\), including relations between various spectra, numerical ranges, and lattices of invariant subspaces. In particular, we show that if\(\tilde T\) has a nontrivial invariant subspace, then so doesT, and we give various applications of our results.
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