For a contraction P and a bounded commutant S of P , we seek a solution X of the operator equation S − S * P = (I − P * P ) 1 2 X(I − P * P )

Abstract. We show that for

A pair of commuting operators (S, P ) defined on a Hilbert space H for which the closed symmetrized bidiscis a spectral set is called a Γ-contraction in the literature. A Γ-contraction (S, P ) is said to be pure if P is a pure contraction, i.e, P * n → 0 strongly as n → ∞. Here we construct a functional model and produce a set of unitary invariants for a pure Γ-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equationand is called the fundamental operator of the Γ-contraction (S, P ). We also discuss some important properties of the fundamental operator.

We find new characterizations for the points in the symmetrized polydisc [Formula: see text], a family of domains associated with the spectral interpolation, defined by [Formula: see text] We introduce a new family of domains which we call the extended symmetrized polydisc [Formula: see text], and define in the following way: [Formula: see text] [Formula: see text] We show that [Formula: see text] for [Formula: see text] and that [Formula: see text] for [Formula: see text]. We first obtain a variety of characterizations for the points in [Formula: see text] and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in [Formula: see text]. Also, we obtain similar characterizations for the points in [Formula: see text], where [Formula: see text]. A set of [Formula: see text] fractional linear transformations plays central role in the entire program. We also show that for [Formula: see text], [Formula: see text] is nonconvex but polynomially convex and is starlike about the origin but not circled.

No abstract

Abstract:We present a set of necessary and su cient conditions that provides a Schwarz lemma for the tetrablock E. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc G . In either case, our results generalize all previous results in this direction for E and G .

No abstract

Abstract. The symmetrized polydisc of dimension three is the setA triple of commuting operators for which Γ 3 is a spectral set is called a Γ 3 -contraction. We show that every Γ 3 -contraction admits a decomposition into a Γ 3 -unitary and a completely nonunitary Γ 3 -contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set Γ 3 and Γ 3 -contractions.Mathematics subject classification (2010): 47A13, 47A15, 47A20, 47A25, 47A45.