The class of complex symmetric operators is not norm closed

Abstract

An operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T element-of script upper B script left-parenthesis script upper H script right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> <mml:mo class="MJX-tex-caligraphic" mathvariant="script" stretchy="false">(</mml:mo> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> <mml:mo class="MJX-tex-caligraphic" mathvariant="script" stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T\in \mathcal {B(H)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is complex symmetric if there exists a conjugate-linear, isometric involution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C colon script upper H long right-arrow script upper H"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ⟶ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">C:\mathcal {H}\longrightarrow \mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper T upper C equals upper T Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>T</mml:mi> <mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">CTC=T^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, a class of complex symmetric operators on finite dimensional Hilbert spaces is constructed. As an application, it is shown that Kakutani’s unilateral weighted shift operator is not complex symmetric; however, it is a norm limit of complex symmetric operators. This gives a negative answer to a question of S. Garcia and W. Wogen: that is, whether or not the class of complex symmetric operators is norm closed.