Multivariable dilation theory on Hilbert spaces

The thesis begins with a survey of the Sz.-Nagy and Foiac{s} dilation theory of contractions on Hilbert spaces. This includes the study of the unilateral shift $S$, which acts as a model operator. Dilation theorems for multiple commuting contractions due to It^{o} and And^{o} are also considered. The extensions of these classical dilation results to three settings due to Popescu are introduced. The first of these extensions are the dilation theorems for row contractions and the model $n$-tuple $(S1,ldots,Sn)$ of left creation operators. We look at the noncommutative disc algebra $mathcalAn$, which is the norm closure of all polynomials in the left creation operators. The next portion concerns the development of a dilation theorem for elements of polyballs which satisfy a positivity condition. The model element in this setting is the family $(Sij)substack1leqileqk 1leqjleqni$ of left creation operators defined on $F2(Hn1)otimescdotsotimesF2(Hnk)$. Finally, a dilation theorem for generalized row contractions is established. This involves the model $n$-tuple $(W1,ldots,Wn)$ of weighted left creation operators.