Speaker: Dr. Shubham Rastogi
Date and time: January 13, 11:30 am
Venue: Ramanujan Hall

Title: On *-regular isometric dilations

Abstract: Every contraction on a Hilbert space has an isometric dilation.
And\^o extended this result to the pairs of commuting contractions. S.
Parrott showed that this dilation result does not extend to an $n$-tuple
of commuting contractions, in general for $n\geq 3.$ However, provided
that the $n$-tuple satisfies Brehmer's positivity condition, the dilation
exists. In fact, S. Brehmer proved that an $n$-tuple  of commuting
contractions satisfies Brehmer's positivity if and only if it admits a
minimal $*$-regular isometric dilation.  Moreover, D. Gasper and N. Suciu
showed that the minimal $*$-regular isometric dilation comprises doubly
commuting isometries. In this talk, we shall see an extension of this
result to a sequence of commuting contractions.

An $n$-tuple of doubly commuting pure isometries can be modeled by the
tuple of multiplication by the co-ordinate functions on a vector-valued
Hardy space over the polydisc. A similar result does not hold true for a
sequence of doubly commuting pure isometries. This brings us to the
question of characterizing a sequence that has the sequence of
multiplication by the co-ordinate functions on a vector-valued Hardy space
over the Hilbert multidisc, as its minimal $*$-regular isometric dilation.
We will address this question in the talk. The talk is based on a work in
progress with B. K. Das.