Does a typical ?_{????}-space contraction have a non-trivial invariant subspace?

Given a Polish topology ? \tau on alttext="script upper B 1 left-parenthesis X right-parenthesis"> B 1 ( X stretchy="false">) encoding="application/x-tex">\mathcal {B}_{1}(X) , the set of all contraction operators alttext="upper equals script l Subscript p"> = ? &gt; mathvariant="normal">?<!-- ? encoding="application/x-tex">1\le p&gt;\infty or c 0"> c 0 encoding="application/x-tex">X=c_0 we prove several results related to following question: does typical T element-of T ?<!-- ? encoding="application/x-tex">T\in \mathcal in Baire Category sense has non-trivial invariant subspace? In other words, is there dense G delta"> G ?<!-- ? encoding="application/x-tex">G_\delta subset-of-or-equal-to right-parenthesis comma tau mathvariant="script">G ?<!-- ? <mml:mo>, G\subseteq (\mathcal {B}_{1}(X),\tau ) such that every G"> G We mostly focus Strong Operator Topology and Strong /> ?<!-- ? </mml:msup> encoding="application/x-tex">^* Topology.