The Kobayashi Distance between two Contractions

Suciu, Ion

doi:10.1007/978-3-0348-8575-1_11

Cited by 3 publications

(7 citation statements)

References 6 publications

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“…Our hyperbolic metric δ is different from the Kobayashi distance δ K on unit ball [B(H) n ] 1 , with respect to the Poincaré distance on D. Indeed, when n = 1, one can show that δ coincides with the Harnack distance introduced by Suciu. In this case, according to [39] (and due to a result from [42]), we have δ(0, A) < δ K (0, A) = 1 2 ln 1 + A 1 − A for certain strict contractions A ∈ B(H) with dim H 2. This also shows that δ is different from the metric for the ball [B(H) n ] 1 , as defined in [11].…”

Section: Proof First We Prove Thatmentioning

confidence: 92%

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Noncommutative hyperbolic geometry on the unit ball ofB(H)n

Popescu

2009

Journal of Functional Analysis

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In this paper we introduce a hyperbolic (Poincaré-Bergman type) distance δ on the noncommutative open ball B(H) n 1 := (X 1 , . . . , X n ) ∈ B(H) n : X 1 X * 1 + · · · + X n X * n 1/2 < 1 , where B(H) is the algebra of all bounded linear operators on a Hilbert space H. It is proved that δ is invariant under the action of the free holomorphic automorphism group of [B(H) n ] 1 , i.e.,). Moreover, we show that the δ-topology and the usual operator norm topology coincide on [B(H) n ] 1 . While the open ball [B(H) n ] 1 is not a complete metric space with respect to the operator norm topology, we prove that [B(H) n ] 1 is a complete metric space with respect to the hyperbolic metric δ.We obtain an explicit formula for δ in terms of the reconstruction operator R X := X * 1 ⊗ R 1 + · · · + X * n ⊗ R n , X := (X 1 , . . . , X n ) ∈ B(H) n 1 , associated with the right creation operators R 1 , . . . , R n on the full Fock space with n generators. In the particular case when H = C, we show that the hyperbolic distance δ coincides with the Poincaré-Bergman ✩ Research supported in part by an NSF grant.

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Section: Proof First We Prove Thatmentioning

confidence: 92%

“…We mention the work of Suciu [38,40,39], Foiaş [8], and Andô, Suciu and Timotin [1] on Harnack parts of contractions and Harnack type distances between two contractions on Hilbert spaces. Some of their results will be recover (with a different proof) in the present paper, in the particular case when n = 1.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative hyperbolic geometry on the unit ball ofB(H)n

Popescu

2009

Journal of Functional Analysis

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“…We consider now the Shmul'yan equivalence of contractions acting between two separable Hilbert spaces E and E ′ . Our first aim is to give a generalization in this context of a result from [18] obtained for contractions acting on same space.…”

Section: Remarks On Shmul'yan and Harnack Equivalencesmentioning

confidence: 99%

“…Both pre-order relations have nice geometric and analytic interpretations. Although these two pre-orders have been around since 1970s and 1980s [1,5,[15][16][17]20,24], their structure is to date not completely understood, and in recent years there has been an increase in interest for this topic [2,7,8,10,12,13,18,19,23].…”

Section: Introductionmentioning

confidence: 99%

“…By[18, Theorem 4.1] and the proof of[19, Theorem 4], any Shmul'yan part is a union of ranges of B(H)-valued contractive analytic functions on D. Thus, Theorem 2.9 asserts that, under some additional commutativity conditions, a contraction T ′ which Harnack dominates a quasi-normal T belongs together with T to the range of an analytic functions on D, which is contained in the Shmul'yan part of T .…”

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confidence: 99%

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

Badea

Suciu

2017

Indagationes Mathematicae

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We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul'yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul'yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul'yan parts coincide. We also discuss an extension, recently considered by S. ter Horst [J. Operator Th. 72(2014), 487-520 ], of the Shmul'yan pre-order from contractions to the operator-valued Schur class of functions. In particular, the Shmul'yan-ter Horst part of a given partial isometry, viewed as a constant Schur class function, is explicitly determined.2010 Mathematics Subject Classification. Primary 47A10, 47A45; Secondary 47A20, 47A35, 47B15.

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Analytic formulas for the hyperbolic distance between two contractions

Suciu¹

1997

Ann. Polon. Math.

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Abstract. In this paper we give some analytic formulas for the hyperbolic (Harnack) distance between two contractions which permit concrete computations in several situations, including the finite-dimensional case. The main consequence of these formulas is the proof of the Schwarz-Pick Lemma. It modifies those given in [13] by the avoidance of a general Schur type formula for contractive analytic functions, more exactly by reducing the case to the more manageable situation when the function takes as values strict contractions.The Harnack equivalence was introduced in [11] and studied in manyThe hyperbolic (Harnack) distance on the Harnack parts was introduced in [15] in a general context of completely positive maps from a subspace of a C * -algebra into B(H). The special case of the contractions was considered in [13], where the Schwarz-Pick Lemma for an operator-valued contractive analytic function defined on the open unit disc in the complex plane was proved.In the present paper we give some analytic formulas for the hyperbolic (Harnack) distance between two contractions (Sections 2, 3) which permit concrete computations in several situations, including the finite-dimensional case. The main consequence of these formulas is the proof of the SchwarzPick Lemma given in Section 4. It modifies those given in [13] by the avoidance of a general Schur type formula for contractive analytic functions, more exactly by reducing the case to the more manageable situation when the function takes strict contractions as values.

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The Kobayashi Distance between two Contractions

Cited by 3 publications

References 6 publications

Noncommutative hyperbolic geometry on the unit ball ofB(H)n

Noncommutative hyperbolic geometry on the unit ball ofB(H)n

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

Analytic formulas for the hyperbolic distance between two contractions

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