Unitary Dilations of Contractions In κκ -Spaces

Abstract

If T is a contraction in a Hilbert space H, H can be decomposed into two orthogonal subspaces H u and H s, H u ⊕ H s , with respect to which T has the matrix representation $$ {\text{T = }}\left( {\begin{array}{*{20}{c}}{{{\text{T}}_{\text{u}}}}&0 \\0&{{{\text{T}}_{\text{s}}}}\end{array}} \right):\left( {\begin{array}{*{20}{c}}{{H_{\text{u}}}} \\{{H_s}}\end{array}} \right) \to \left( {\begin{array}{*{20}{c}}{{H_{\text{u}}}} \\{{H_s}}\end{array}} \right), $$ (0.1) where Tu is the unitary part and Ts the simple part of T, see [12] and [15]. If U in the Hilbert space \( \tilde H \), containing H as a subspace, is the (uniquely determined) minimal unitary dilation of T, then U has the representation $$ {\text{U = }}\left( {\begin{array}{*{20}{c}} {{{\text{T}}_{{\text{u}}}}} 0 0 \end{array} {\text{ }}\begin{array}{*{20}{c}} 0 {{{\text{T}}_{{\text{s}}}}} {{{\text{G}}_{{\text{s}}}}} \end{array} {\text{ }}\left. {\begin{array}{*{20}{c}} 0 {{{\text{F}}_{{\text{s}}}}} {{{\text{H}}_{{\text{s}}}}} \end{array} } \right)} \right.:\left( {\begin{array}{*{20}{c}} {{H_{{\text{u}}}}} {{H_{{\text{s}}}}} {{H_{{\text{1}}}}} \end{array} } \right) \to \left( {\begin{array}{*{20}{c}} {{H_{{\text{u}}}}} {{H_{{\text{s}}}}} {{H_{{\text{1}}}}} \end{array} } \right), $$ (0.2) where H 1 =~ H⊖H and $$ {{\text{U}}_{{\text{s}}}} = \left( {\begin{array}{*{20}{c}} {{{\text{T}}_{{\text{s}}}}} {{{\text{G}}_{{\text{s}}}}} \end{array} } \right.{\text{ }}\left. {\begin{array}{*{20}{c}} {{{\text{F}}_{{\text{s}}}}} {{{\text{H}}_{{\text{s}}}}} \end{array} } \right):\left( {\begin{array}{*{20}{c}} {{H_{{\text{s}}}}} {{H_{1}}} \end{array} } \right) \to \left( {\begin{array}{*{20}{c}} {{H_{{\text{s}}}}} {{H_{1}}} \end{array} } \right) $$ is the minimal unitary dilation of Ts. Now, if T is a contraction in a Pontryagin space Π, it also has a uniquely determined minimal unitary dilation (see [14]) and the question came up whether or not the above matrix representations still hold when H is replaced by Π. We show in Sections 1 and 2 that this is not the case, due to the fact that the maximal subspace U of Π on which T acts like a unitary operator may be degenerated. If it is, then the representation of T analogous to (0.1) is given by a 4×4 matrix (see Theorem 1.1) and we show in Section 2 that in order to express U in terms of Tu and the minimal unitary dilation of Ts as in (0.2), a 6×6 matrix repesentation is required. If the isotropic part of U is trivial, these representations reduce to ones of the form (0.1) and (0.2). In [15] it is shown that the spectrum of the minimal unitary dilation U of a simple contraction T in a Hilbert space H is absolutely continuous with respect to the Lebesque measure on the unit circle in C.