[Paper Review] Hypercontractions and factorizations of multipliers in one and several variables

Monojit Bhattacharjee, B. Krishna Das+1 more|arXiv (Cornell University)|Dec 19, 2018

Holomorphic and Operator Theory19 references4 citations

TL;DR

This paper introduces characteristic functions for commuting tuples of hypercontractions in one and several variables, generalizing Sz.-Nagy and Foias' theory. It establishes a canonical factorization of these characteristic functions via universal multipliers and transfer functions, proving they are complete unitary invariants and fully determine joint invariant subspaces.

ABSTRACT

We introduce the notion of characteristic functions for commuting tuples of hypercontractions on Hilbert spaces, as a generalization of the notion of Sz.-Nagy and Foias characteristic functions of contractions. We present an explicit method to compute characteristic functions of hypercontractions and relate characteristic functions by means of the factors of Schur-Agler class of functions and universal multipliers on the unit ball in $\mathbb{C}^n$. We also offer some factorization properties of multipliers. Characteristic functions of hypercontrctions are complete unitary invariant. The Drury-Arveson space and the weighted Bergman spaces on the unit ball continues to play a significant role in our consideration. Our results are new even in the special case of single hypercontractions.

Motivation & Objective

• To generalize the Sz.-Nagy and Foias characteristic function theory to commuting tuples of hypercontractions in several variables.
• To define a canonical characteristic triple for pure $m$-hypercontractions and derive their characteristic functions as operator-valued analytic functions on the unit ball.
• To establish that characteristic functions are complete unitary invariants for hypercontractions, uniquely determining joint invariant subspaces.
• To provide a factorization of characteristic functions into universal multipliers and transfer functions, extending classical model theory to hypercontractions.
• To characterize wandering subspaces of joint shift-invariant subspaces using the factorization of contractive multipliers.

Proposed method

• Introduce a characteristic triple $(\mathcal{E}, B, D)$ for a pure $m$-hypercontraction $T$, consisting of a Hilbert space $\mathcal{E}$ and bounded operators $B, D$.
• Define the characteristic function $\Theta_T(\bm{z})$ as an operator-valued analytic function on $\mathbb{B}^n$ derived from the characteristic triple.
• Construct a canonical transfer function $\tilde{\Phi}_{T,m}(\bm{z})$ using a unitary colligation $U = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$, with $\tilde{\Phi}_{T,m}(\bm{z}) = D + C(I - ZA)^{-1}ZB$.
• Prove that the purely contractive part of $\tilde{\Phi}_{T,m}$, restricted to $(\ker B)^\perp$, equals the characteristic function $\Theta_T(\bm{z})$ via unitary and isometric intertwining.
• Establish a general factorization theorem for contractive multipliers from vector-valued Drury-Arveson spaces to reproducing kernel Hilbert spaces on $\mathbb{B}^n$.
• Use the factorization to parametrize wandering subspaces of joint shift-invariant subspaces in terms of factors of the characteristic function.

Experimental results

Research questions

• RQ1 Can the characteristic function of a pure $m$-hypercontraction be canonically defined in several variables, generalizing the single-contraction case?
• RQ2 How are the characteristic functions of hypercontractions related to Schur-Agler class functions and universal multipliers on the unit ball?
• RQ3 To what extent do characteristic functions serve as complete unitary invariants for commuting tuples of hypercontractions?
• RQ4 Can the characteristic function of a hypercontraction be factorized into a universal multiplier and a transfer function?
• RQ5 How do the factors of the characteristic function determine the joint invariant subspaces of the hypercontraction?

Key findings

• The characteristic function $\Theta_T(\bm{z})$ of a pure $m$-hypercontraction is a complete unitary invariant, uniquely determining the operator up to unitary equivalence.
• The characteristic function $\Theta_T(\bm{z})$ is canonically represented as $Y\tilde{\Phi}_{T,m}(\bm{z})X$, where $\tilde{\Phi}_{T,m}$ is the transfer function from a characteristic triple and $X, Y$ are unitary and isometric maps.
• The purely contractive part of the transfer function $\tilde{\Phi}_{T,m}(\bm{z})$ coincides with the characteristic function $\Theta_T(\bm{z})$, establishing a link to classical model theory.
• The joint invariant subspaces of a pure $m$-hypercontraction are completely determined by the factors of its characteristic function.
• A general factorization theorem for contractive multipliers on $\mathbb{B}^n$ is established, enabling parametrization of wandering subspaces of shift-invariant subspaces.
• The results are new even in the case of single hypercontractions, extending the classical Sz.-Nagy–Foias theory to the hypercontraction setting in several variables.