Necessary Conditions for E(3; 3; 1, 1, 1)-Isometric Dilation, E(3; 2; 1, 2)-Isometric Dilation and P-Isometric Dilation
Abstract
A fundamental theorem of Sz.-Nagy states that a contraction T on a Hilbert space can be dilated to an isometry V. A more multivariable context of recent significance for these concepts involves substituting the unit disk with E(3; 3; 1, 1, 1), E(3; 2; 1, 2), and pentablock. We demonstrate the necessary conditions for the existence of E(3; 3; 1, 1, 1)-isometric dilation, E(3; 2; 1, 2)-isometric dilation and pentablock-isometric dilation. We construct a class of E(3; 3; 1, 1, 1)-contractions and E(3; 2; 1, 2)-contractions that are always dilate . We create an example of a E(3; 3; 1, 1, 1)-contraction that has a E(3; 3; 1, 1, 1)-isometric dilation such that [F7-i*, Fj] [F7-j*, Fi] for some i,j with 1≤ i ,j≤ 6, where Fi and F7-i, 1≤ i ≤ 6 are the fundamental operators of E(3; 3; 1, 1, 1)-contraction T=(T1, …, T7). We also produce an example of a E(3; 2; 1, 2)-contraction that has a E(3; 2; 1, 2)-isometric dilation by which [G*1, G1] ≠ [G*2, G2]~ and ~[2G*2, 2G2] ≠ [2G*1, 2G1], where G1, 2G2, 2G1, G2 are the fundamental operators of S. As a result, the set of sufficient conditions for the existence of a E(3; 3; 1, 1, 1)-isometric dilation and E(3; 2; 1; 2) -isometric dilations presented in Theorem conddilation and Theorem condilation1, respectively, are not generally necessary. We construct explicit E(3; 3; 1, 1, 1) -isometric, E(3; 2; 1; 2) -isometric dilations and P-isometric dilation of E(3; 3; 1, 1, 1)-contraction, E(3; 2; 1; 2)-contraction and P-contraction, respectively.
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