Analytic formulas for the hyperbolic distance between two contractions

Ion Suciu

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 239-252
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Ion Suciu. "Analytic formulas for the hyperbolic distance between two contractions." Annales Polonici Mathematici 66.1 (1997): 239-252. <http://eudml.org/doc/269949>.

@article{IonSuciu1997,
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.},
author = {Ion Suciu},
journal = {Annales Polonici Mathematici},
keywords = {Harnack parts; hyperbolic distance; operator Schwarz-Pick Lemma; hyperbolic distance between two contractions; Schwarz-Pick Lemma; avoidance of a general Schur type formula for contractive analytic functions},
language = {eng},
number = {1},
pages = {239-252},
title = {Analytic formulas for the hyperbolic distance between two contractions},
url = {http://eudml.org/doc/269949},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Ion Suciu
TI - Analytic formulas for the hyperbolic distance between two contractions
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 239
EP - 252
AB - 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.
LA - eng
KW - Harnack parts; hyperbolic distance; operator Schwarz-Pick Lemma; hyperbolic distance between two contractions; Schwarz-Pick Lemma; avoidance of a general Schur type formula for contractive analytic functions
UR - http://eudml.org/doc/269949
ER -

References

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  1. [1] T. Ando, I. Suciu and D. Timotin, Characterization of some Harnack parts of contractions, J. Operator Theory 2 (1979), 233-245. Zbl0431.47005
  2. [2] S. Dineen, The Schwarz Lemma, Clarendon Press, Oxford, 1989. Zbl0708.46046
  3. [3] C. Foiaş, On Harnack parts of contractions, Rev. Roumaine Math. Pures Appl. 19 (1974), 314-318. Zbl0287.47007
  4. [4] C. Foiaş and A. E. Frazho, The Commutant Lifting Aproach to Interpolation Problems, Oper. Theory Adv. Appl. 44, Birkhäuser, Basel, 1990. 
  5. [5] V. A. Khatskevich, Yu. L. Shmul'yan and V. S. Shul'man, Equivalent contractions, Dokl. Akad. Nauk SSSR 278 (1) (1984), 47-49 (in Russian); English transl.: Soviet Math. Dokl. 30 (2) (1984), 338-340. 
  6. [6] V. A. Khatskevich, Yu. L. Shmul'yan and V. S. Shul'man, Pre-orders and equivalences in the operator ball, Sibirsk. Mat. Zh. 32 (3) (1991) (in Russian); English transl.: Siberian Math. J. 32 (3) (1991), 496-506. 
  7. [7] W. Mlak, Hilbert Spaces and Operator Theory, PWN-Kluwer, Warszawa-Dordrecht, 1991. 
  8. [8] G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23. Zbl45.0642.01
  9. [9] H. A. Schwarz, Zur Theorie der Abbildung, in: Gesammelte Mathematische Abhandlungen, Band II, Springer, Berlin, 1890, 108-132. 
  10. [10] S. Stoilow, Theory of Functions of a Complex Variable, I, II, Editura Academiei, Bucureşti, 1954, 1958 (in Romanian). 
  11. [11] I. Suciu, Harnack inequalities for a functional calculus, in: Hilbert Space Operators and Operator Algebras (Proc. Internat. Conf., Tihany, 1970), Colloq. Math. Soc. János Bolyai 5, North-Holland, Amsterdam, 1972, 449-511. 
  12. [12] I. Suciu, Analytic relations between functional models for contractions, Acta Sci. Math. (Szeged) 33 (1973), 359-365. Zbl0265.47011
  13. [13] I. Suciu, Schwarz Lemma for operator contractive analytic functions, preprint IMAR No. 5, 1992. 
  14. [14] I. Suciu, The Kobayashi distance between two contractions, in: Oper. Theory Adv. Appl. 61, Birkhäuser, Basel, 1993, 189-200. Zbl0796.47008
  15. [15] I. Suciu and I. Valuşescu, On the hyperbolic metric on Harnack parts, Studia Math. 55 (1976), 97-109. Zbl0328.46074
  16. [16] N. Suciu, On Harnack ordering of contractions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 467-471. Zbl0442.47006
  17. [17] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland-American Elsevier-Akademiai Kiadó, Amsterdam-New York-Budapest, 1970. 

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