Location
Brown Hall 100
Start Date
7-22-2016 11:40 AM
End Date
22-7-2016 12:30 PM
Description
The starting point for the Nagy-Foias model for a contractive operator $T$ on Hilbert space is Sz.-Nagy's observation that $T$ has a canonical minimal unitary dilation to a larger Hilbert space. For a {\em pair} $T=(T_1,T_2)$ of commuting contractions, Ando's theorem asserts that there exist commuting unitary dilations of $T$ to larger Hilbert spaces, and one might aspire to extend the Nagy-Foias model to such operator pairs. However, the dilations provided by Ando's theorem are far from being canonical, and this fact appears to rule out a good model theory for $T$. It has recently been shown that nevertheless there {\em is} such a model theory, though it requires a slight shift in perspective. One focuses on the commuting pair $(S,P)$, where $S=T_1+T_2,\; P=T_1T_2$. The operator pair $(S,P)$ is a {\em $\Gamma$-contraction}, which means that the set \[ \Gamma =\{(z+w,zw): |z|\leq 1, \, |w|\leq 1\} \] is a spectral set for $(S,P)$. One constructs a canonical dilation, and thereafter a functional model, not for the individual operators $T_1, T_2$, but for the pair $(S,P)$. The model parallels closely the original Nagy-Foias model. This line of investigation was begun by Agler and Young, and successfully developed and brought to a conclusion by members of the Indian school of operator theory.
A Nagy-Foias model for commuting pairs of contractions
Brown Hall 100
The starting point for the Nagy-Foias model for a contractive operator $T$ on Hilbert space is Sz.-Nagy's observation that $T$ has a canonical minimal unitary dilation to a larger Hilbert space. For a {\em pair} $T=(T_1,T_2)$ of commuting contractions, Ando's theorem asserts that there exist commuting unitary dilations of $T$ to larger Hilbert spaces, and one might aspire to extend the Nagy-Foias model to such operator pairs. However, the dilations provided by Ando's theorem are far from being canonical, and this fact appears to rule out a good model theory for $T$. It has recently been shown that nevertheless there {\em is} such a model theory, though it requires a slight shift in perspective. One focuses on the commuting pair $(S,P)$, where $S=T_1+T_2,\; P=T_1T_2$. The operator pair $(S,P)$ is a {\em $\Gamma$-contraction}, which means that the set \[ \Gamma =\{(z+w,zw): |z|\leq 1, \, |w|\leq 1\} \] is a spectral set for $(S,P)$. One constructs a canonical dilation, and thereafter a functional model, not for the individual operators $T_1, T_2$, but for the pair $(S,P)$. The model parallels closely the original Nagy-Foias model. This line of investigation was begun by Agler and Young, and successfully developed and brought to a conclusion by members of the Indian school of operator theory.