Singular values of iterated Aluthge transforms of compact operators

Abstract

Yamazaki proved that the norms of the iterated Aluthge transforms of an operator converge to its spectral radius, and Ando showed that for matrices the singular values of the iterates converge to the moduli of the eigenvalues. We prove that the latter holds for every compact operator on a Hilbert space and every generalized Aluthge transform $Δ_t$ with $0<t<1$: for every $j$, the $j$-th singular value of the $n$-th iterate converges to $|λ_j(T)|$, and the products of the $k$ largest singular values, as well as the Ky Fan norms, decrease monotonically to their limits. This is the Aluthge counterpart of the Yamamoto--Davis theorem. We also determine the limit in the Schatten $p$-norm whenever $T$ belongs to a strictly smaller Schatten class.