The mean transform and the mean limit of an operator

Chabbabi, Fadil; Curto, Raúl; Mbekhta, Mostafa

doi:10.1090/proc/14277

The mean transform and the mean limit of an operator
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by Fadil Chabbabi, Raúl E. Curto and Mostafa Mbekhta PDF

Proc. Amer. Math. Soc. 147 (2019), 1119-1133 Request permission

Abstract:

Let $T\in \mathcal {B}(\mathcal {H})$ be a bounded linear operator on a Hilbert space $\mathcal {H}$, and let $T \equiv V|T|$ be the polar decomposition of $T$. The mean transform of $T$ is defined by $\widehat {T}:=\frac {1}{2}(V|T|+|T|V)$. In this paper we study the iterates of the mean transform and we define the mean limit of an operator as the limit (in the operator norm) of those iterates. We obtain new estimates for the numerical range and numerical radius of the mean transform in terms of the original operator. For the special class of unilateral weighted shifts we describe the precise relationship between the spectral radius and the mean limit, and obtain some sharp estimates.

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