Similarity of C1: Operators and the Hyperinvariant Subspace Problem

In the present paper, we first show that the existence of the solutions of the operator equation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"><a:msup><a:mrow><a:mi>S</a:mi></a:mrow><a:mi>∗</a:mi></a:msup><a:mtext>XT</a:mtext><a:mo>=</a:mo><a:mi>X</a:mi></a:math> is related to the similarity of operators of class <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"><c:msub><c:mrow><c:mi>C</c:mi></c:mrow><c:mrow><c:mn>1</c:mn><c:mo>.</c:mo></c:mrow></c:msub></c:math> , and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"><e:mtext>ran</e:mtext><e:mi>φ</e:mi><e:mfenced open="(" close=")" separators="|"><e:mrow><e:mi>T</e:mi></e:mrow></e:mfenced></e:math> for some singular inner functions <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M5"><j:mi>φ</j:mi></j:math> . As an application, we prove that every <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M6"><l:msub><l:mrow><l:mi>C</l:mi></l:mrow><l:mrow><l:mn>10</l:mn></l:mrow></l:msub></l:math> -quasinormal operator and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M7"><n:msub><n:mrow><n:mi>C</n:mi></n:mrow><n:mrow><n:mn>10</n:mn></n:mrow></n:msub></n:math> -centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.