Contractions on \(\Pi_ k\) spaces (Q1178396)

Let \(\Pi_ k\) be a Pontrjagin space, \((\cdot,\cdot)\) the indefinite inner product on \(\Pi_ k\). A linear bounded operator \(T\) is called a contraction if \((Tx,Tx)\leq(x,x)\) for all \(x\in\Pi_ k\). In this brief report the authors announce some results, without proof, on contractions. The most interesting result is one about the triangle models of contractions. Suppose \(T\) is a contraction on \(\Pi_ k\), then there is a standard decomposition \(\Pi_ k=N\oplus\{Z+Z^*\}\oplus P\) of \(\Pi_ k\) such that \[ T=\begin{pmatrix} S &F &G &B\\ &T_ N &T_ 1 &C\\ &&T_ P &D\\ &&&S^{\ast-1} \end{pmatrix} \begin{matrix} Z\\ N\\ P\\ Z^* \end{matrix} \] with \(S\) injective on \(Z\), \(B={1\over2}S(C^*C-D^*D+2Q)\), \(Q+Q^*\leq 0\) and \(\begin{pmatrix} T_ N &T_ 1\\ &T_ P \end{pmatrix}\) a contraction on \(N\oplus P\). It is also announced that every strongly continuous semi- group of contractions on \(\Pi_ k\) has a \(J\)-unitary dilation strongly continuous semi-group on some \(\Pi_ k'\supseteq\Pi_ k\).