Polydisc contractions and regular dilations

A set of positivity conditions (obtained by Brehmer) characterizes the existence of a regular unitary dilation for a commuting multioperator. A certain positive operator, denoted by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J 0"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">J_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and introduced by Curto and Vasilescu in the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -contractions, can be still defined if our commuting multioperator is just a polydisc contraction. We present some alternative formulas for the description of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J 0"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">J_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We give sufficient conditions (related to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J 0"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">J_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) on a polydisc contraction in order to ensure the existence of regular isometric dilations and discuss the connection with the model of Curto - Vasilescu and Timotin. We prove that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a regular isometric dilation, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J 0 equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">J_0=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polydisc isometry if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an extension in which its double commuting parts have at least one unitary operator. We also show that a multicontraction and a polydisc contraction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have a backwards multishift extension if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript 0 dot"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> <mml:mo> ⋅ </mml:mo> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">C_{0\cdot }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J 0 equals 1 Subscript script upper H"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">J_0=1_\mathscr {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main result of this paper describes the parts in the Curto-Vasilescu-Timotin model when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 Subscript script upper H Baseline minus upper T Subscript i Superscript asterisk Baseline upper T Subscript i Baseline equals upper J 0 minus upper T Subscript i Superscript asterisk Baseline upper J 0 upper T Subscript i"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> </mml:msub> <mml:mo> − </mml:mo> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub