ACTA issues

A representation of unitary operators in ${\Pi}_{\kappa}$ in terms of isometric operators and orthogonal projections

Muhamed Borogovac

Acta Sci. Math. (Szeged) 77:3-4(2011), 551-565

Abstract. Let $\Pi_{\kappa }$ be a Pontryagin space with decomposition $\Pi_{\kappa }=\Pi_{+}(+)\Pi_{-}$, $\kappa =\dim\Pi _{+}< \infty $. Let $U$ be a unitary operator in $\Pi_{\kappa }$ and let $U= \left[{A B\atop C D}\right ]$ be its matrix representation that corresponds to the given decomposition of $\Pi_{\kappa }$. In this note operators $A$, $B$, $C$, and $D$ are given in terms of isometric operators and orthogonal projections in a way that those expressions are necessary and sufficient conditions for the operator $U$ to be unitary. The results are more specific and intuitive than the results from the last chapter of [2]. The obtained representation of $U$ is applied to study operator $T$ that has $\kappa $-dimensional positive invariant subspace $J_{+}$ and allows a J-polar decomposition. The radius of the spectrum $\sigma(T\mid J_{+}) $ is estimated.

AMS Subject Classification (1991): 47B50, 46C20

Keyword(s): Pontryagin space, unitary operator in $\Pi_{\kappa }$, polar decomposition

Received April 16, 2010, and in revised form May 6, 2011. (Registered under 29/2010.)