# Journal of Operator Theory

Volume 72, Issue 2, Fall 2014 pp. 529-547.

A canonical decomposition of complex symmetric
operators

**Authors**:
Kunyu Guo (1) and Sen Zhu (2)

**Author institution:** (1) School of Mathematical Sciences, Fudan
University,\break Shanghai, 200433, P.R. China

(2) Department of Mathematics, Jilin University, Changchun, 130012, P.R.
China

**Summary: ** An operator $T$ on a complex Hilbert space
$\mathcal{H}$ is said to be complex
symmetric if there exists a conjugate-linear, isometric involution
$C:\mathcal{H}\rightarrow\mathcal{H}$ so that $CTC=T^*$. In this paper, we
obtain a canonical decomposition of complex symmetric operators. This result
decomposes general complex symmetric operators into direct sums of three
kinds of elementary ones.

**DOI: **http://dx.doi.org/10.7900/jot.2013aug15.2007

**Keywords: ** complex symmetric operator, transpose, UET operator,
canonical decomposition, irreducible operator, completely reducible
operator, Toeplitz operator

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