A 𝐶*-algebra approach to complex symmetric operators

Guo, Kunyu; Ji, Youqing; Zhu, Sen

doi:10.1090/S0002-9947-2015-06215-1

A $C^*$-algebra approach to complex symmetric operators
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by Kunyu Guo, Youqing Ji and Sen Zhu PDF

Trans. Amer. Math. Soc. 367 (2015), 6903-6942 Request permission

Abstract:

In this paper, certain connections between complex symmetric operators and anti-automorphisms of singly generated $C^*$-algebras are established. This provides a $C^*$-algebra approach to the norm closure problem for complex symmetric operators. For $T\in \mathcal {B(H)}$ satisfying $C^*(T)\cap \mathcal {K(H)}=\{0\}$, we give several characterizations for $T$ to be a norm limit of complex symmetric operators. As applications, we give concrete characterizations for weighted shifts with nonzero weights to be norm limits of complex symmetric operators. In particular, we prove a conjecture of Garcia and Poore. On the other hand, it is proved that an essentially normal operator is a norm limit of complex symmetric operators if and only if it is complex symmetric. We obtain a canonical decomposition for essentially normal operators which are complex symmetric.

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