Analytic model of doubly commuting contractions

Abstract

An n-tuple (n ≥ 2), T = (T 1 , . . . , T n ), of commuting bounded linear operators on a Hilbert space H is doubly commuting if T i T * j = T * j T i for all 1 ≤ i < j ≤ n. If in addition, each T i ∈ C •0 , then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple T can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H 2 D T * (D n ). As a consequence of the dilation theorem, we prove that there exists a closed subspace S T of the formwhere) is either a one variable inner function in z i , or the zero function.

Doubly commuting mixed invariant subspaces in the polydisc

“…We record the following known result on the analytic model for doubly commuting tuple (see [5]). Let T = (T 1 , .…”

Operator positivity and analytic models of commuting tuples of operators

“…. , 1) we recover the functional model for doubly commuting tuples of pure contractions [9]. Moreover, the methods used here are different from those used in [9].…”

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions

“…Under some additional conditions, the above graceful analytic model for single contraction of class C •0 can be generalized to the situation of a commuting finite-tuple (T 1 , • • • , T n ) of C •0contractions. That is to say, this tuple (T 1 , • • • , T n ) has a dilation to the tuple (M ζ1 , • • • , M ζn ) of coordinate multiplication operators on a E-valued analytic function space H E = H ⊗ E with H consisting of holomorphic functions over some domain in C n [13,28,33,2,9]. The particular case that the tuple being doubly commuting is rather interesting since in this case the function space H is exactly the Hardy space H 2 (D n ) over the n-polydisc, and the underlying space E is the defect space of the tuple (T * 1 , • • • , T * n ).…”

“…The particular case that the tuple being doubly commuting is rather interesting since in this case the function space H is exactly the Hardy space H 2 (D n ) over the n-polydisc, and the underlying space E is the defect space of the tuple (T * 1 , • • • , T * n ). Moreover, we can require this dilation to be minimal and regular [9]. Recall that two operators T , S are said to be doubly commuting if T S = ST and T * S = ST * , and a tuple or a sequence of operators is said to be doubly commuting if any pair of operators in it are doubly commuting.…”

“…The key ingredient in the dilation theory for doubly commuting finite-tuples of C •0 -contractions is the following norm identity [9]:…”