Noncommutative hyperbolic geometry on the unit ball of B (H)n

In this paper we introduce a hyperbolic distance $\delta$ on the noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $\delta$ is invariant under the action of the group $Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\cH)^n]_1$, and show that the $\delta$-topology and the usual operator norm topology coincide on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric space with respect to the hyperbolic metric and obtained an explicit formula for $\delta$ in terms of the reconstruction operator. A Schwarz-Pick lemma for bounded free holomorphic functions on $[B(H)^n]_1$, with respect to the hyperbolic metric, is also obtained.

Noncommutative hyperbolic geometry on the unit ball of $B(H)^n$

https://www.math.uni-bielefeld.de/documenta/vol-14/21.pdf

Hyperbolic geometry on noncommutative polyballs

/pdf/analytic-formulas-for-the-hyperbolic-distance-between-two-3z1rh2johi.pdf

Analytic formulas for the hyperbolic distance between two contractions

Harnack and Shmul’yan pre-order relations for Hilbert space contractions

The Kobayashi pseudodistance can be defined on each complex Banach manifold. Using contractive analytic functions from the unit disk into B(H) we can define the Kobayashi pseudodistance on the closed unit ball B 1 of B(H), where B(H) is the algebra of all bounded linear operators on the Hubert space H. The main result of the paper asserts the fact the Kobayashi pseudodistance is a true distance on the Shmuiyan parts of B 1. Some connections between Pick condition for the two-nodes Nevanlinna-Pick interpolation problem and the Kobayashi distance are established.

The Kobayashi Distance between two Contractions

In this paper we present a geometric approach for studying positive definite sequence of analytic Toeplitz operators following the lines of the general framework of operator extrapolation theory based on choice sequences (cf.[1], [3], [6], [11]). The specific problem here appears is when the positive definite sequence (Tn)n > 1 produced by a choice sequence (Rn)n>i initiated on H2is a sequence of analytic Toeplitz operators. Using special matrix representation based on the choice sequence of the minimal isometric dilation [K, V] of (Tn)n≥1, we prove, in Section 2, that T, is analytic Toeplitz operator for any n if and only if there exists an isometry U on K which commutes with V such that H2, as the subspace of K, reduces U to the shift operator S on H2(Theorem 1). This connects our problem with the topics of Ando dilations (cf.[2], [7], [8]). In Section 3 we write some recurrent formulas appearing in this theory under the specific forms of our context. Section 4 deals with the maximum entropy truncable sequences i.e. the sequences (Tn)n ≥1 of analytic Toeplitz operators whose choice sequence (Rn)n≥1 has the property that for each p ≥ 1, the truncated choice sequence {R1, R2,…, R t 0, 0,…} still produces positive definite sequence of analytic Toeplitz operators. We proved that this happens if and only if, in the matricial form based on the choice sequence, the isometry U is diagonal (Theorem 5). Then we are able to give an algorithm to produce, by a system of free parameters, the set of all maximum entropy truncable positive definite sequences of analytic Toeplitz operators. We end with some considerations concerning a special moment problem on Z2 connected with this results (Sectin 5).

Maximum Entropy Truncable Positive Definite Sequences of Analytic Toeplitz Operators

A rectangular n × m-matrix of complex numbers can be extended to a full quadrant of the Fourier coefficients of a representing measure on the bitorus only in very special conditions. The aim of this paper is to describe the nature of the obstructions which appear when we try to make this extension line by line by using the method of choice sequences from the operator extrapolation theory. The obstructions are described by a sequence of Hankel operators which appear in the recursive construction of an Ando dilation of the pair of the null contractions on C, canonically associated to the representing measure.

Ion Suciu

Papers

The Kobayashi Distance between two Contractions

Maximum Entropy Truncable Positive Definite Sequences of Analytic Toeplitz Operators

A Trigonometric Moment Problem for Z2