Pairs of commuting contractions: Ando lifts and functional models
Authors: Ball Joseph A. 
Published: November 2022

The Wold decomposition for an isometric Hilbert-space operator leads to a functional model for such an operator: the shift operator on a vectorial Hardy space direct sum with a unitary operator for which there is a good spectral theory (complete unitary invariants consisting of the scalar spectral measure together with multiplicity function on the unit circle). Sz.-Nagy and Foias (late 1960s) showed how to construct a functional model for a completely non-unitary contraction operator T by understanding how to embed such a functional-model space into the Wold-decomposition functional-model for the single product isometry V=V_1 V_2.

This story extends to the setting of commuting pairs of isometries and commuting pairs of contractions as follows. Berger-Coburn-Lebow (1978) found a functional model for a commuting pair of isometries (V_1, V_2) by using the Wold-decomposition functional-model for the single product isometry V = V_1 V_2 as the ambient functional-model space. They also found a complete unitary invariant (a “BCL tuple” consisting of a Hilbert coefficient space together with a unitary and projection operator on this space) from which one can construct the functional model. In the case where the product isometry is pure, V is multiplication by the coordinate function z while V_1 and V_2 are equal to multiplication by certain operator pencils with coefficients coming from the BCL tuple. In this talk we discuss the next step: finding a functional model for a commutative pair of contraction operators (T_1, T_2) by seeing how to embed such a space into the BCL functional model for its minimal Ando isometric lift (V_1, V_2). This amounts to finding the models for the commutative operator pair (T_1, T_2) as acting on the Sz.-Nagy-Foias functional-model space for the single product operator T = T_1 T_2.

Our approach to the Sz.-Nagy-Foias model was inspired by the approach of Douglas (1968). A complete invariant consists of (i) the Sz.-Nagy-Foias characteristic function for the single product contraction operator T, (ii) the BCL tuple for the Ando lift (V_1, V_2), and (iii) an embedding operator Λ of the defect space of T^* into the defect space of V^* = V_1^* V_2^* which again is subject to some additional operator equations. Part of this construction is a functional model for any Ando lift (V_1, V_2) for the commutative contractive pair (T_1, T_2). Ando lifts are classified by a collection of spaces and operators which we call Ando tuples (the complete unitary invariant for (T_1, T_2) but with the characteristic function Θ_T for T = T_1 T_2 dropped).

There is a special class of Ando lifts for which it can be proved that the additional operator equations are automatically satisfied, leading to a new functional-model proof of the Ando dilation theory itself. Parallel results hold for commuting contractive d-tuples.

The fact that the Ando dilation theorem does not extend in general to commutative contractive d-tuples once d >2 is explained by the fact that the auxiliary supplementary operator equations fail to hold in general. This is joint work with Haripada Sau of the Indian Institute of Technology Guwahati.