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- W2785130070 abstract "One of the most important results in operator theory is Ando's cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Schaffer cite{sfr} and Douglas cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an Ando dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of Ando's dilation theorem by giving both Schaffer-type and Douglas-type explicit constructions of an Ando dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction $T$ into the product of two commuting contractions. Unlike the one-variable case, two minimal Ando dilations need not be unitarily equivalent. However, we show that the compressions of the two Ando dilations constructed in this paper to the minimal dilation spaces of the contraction $T_1T_2$, are unitarily equivalent. In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}to 0$ strongly, an Ando dilation was constructed recently in cite{D-S-S}, which, as this paper will show, is a corollary to the Douglas-type construction. We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a pure contraction." @default.
- W2785130070 created "2018-02-02" @default.
- W2785130070 creator A5029743170 @default.
- W2785130070 date "2017-10-31" @default.
- W2785130070 modified "2023-09-27" @default.
- W2785130070 title "And^o dilations for a pair of commuting contractions: two explicit constructions and functional models." @default.
- W2785130070 hasPublicationYear "2017" @default.
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