Isometric operators on \(\Pi_ K\) spaces (Q1310509)

The authors' summary: In this paper we give definitions of generalized triangle model, generalized Wold decomposition, Wold decomposition, and regular Wold decomposition of an isometric operator on Pontryagin space \(\Pi_ K\). In the first section we obtain all forms of \(U\)-dilations of an isometric operator on \(\Pi_ K\) under any generalized standard decomposition. In the second section we obtain two results that any isometric operator on \(\Pi_ K\) has Wold decomposition and the unilateral parts of generalized Wold decompositions for an isometric operator on \(\Pi_ K\) are unitarily equivalent one to another. In the last section we get a necessary and sufficient condition under which an isometric operator on \(\Pi_ K\) has regular Wold decomposition, and give a class of isometric operators on \(\Pi_ \ell\) which do not have regular Wold decompositions. Our necessary and sufficient condition is simpler than B. W. McEnnis' one.