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Schur analysis over the unit spectral ball (2402.10718v1)

Published 16 Feb 2024 in math.FA, math-ph, and math.MP

Abstract: We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers.

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References (56)
  1. D. Alpay. An advanced complex analysis problem book. Topological Vector Spaces, Functional Analysis, and Hilbert spaces of analytic functions. Birkhäuser/Springer Basel AG, Basel, 2015.
  2. Spectral factorization in the non-stationary wiener algebra, 2003.
  3. D. Alpay and V. Bolotnikov. Two-sided interpolation for matrix functions with entries in the Hardy space. Linear Algebra Appl., 223/224:31–56, 1995. Special issue honoring Miroslav Fiedler and Vlastimil Pták.
  4. D. Alpay and V. Bolotnikov. On tangential interpolation in reproducing kernel Hilbert space modules and applications. In H. Dym, B. Fritzsche, V. Katsnelson, and B. Kirstein, editors, Topics in interpolation theory, volume 95 of Operator Theory: Advances and Applications, pages 37–68. Birkhäuser Verlag, Basel, 1997.
  5. Brune sections in the non-stationary case. Linear Algebra Appl., 343/344:389–418, 2002. Special issue on structured and infinite systems of linear equations.
  6. Nonstationary analogues of the Herglotz representation theorem for unbounded operators. Arch. Math. (Basel), 78(6):465–474, 2002.
  7. Commutators on Fock spaces. J. Math. Phys., 64:042102–21pages, 2023.
  8. D. Alpay and I. Cho. Certain invertible operator-block matrices induced by C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras and scaled hypercomplex numbers. Res. Math. Sci., 10(4):Paper No. 47, 44, 2023.
  9. Complementation and de Branges Rovnyak spaces: the quaternionic case (tentative title). In preparation.
  10. Schur functions and their realizations in the slice hyperholomorphic setting. Integral Equations and Operator Theory, 72:253–289, 2012.
  11. Krein–Langer Factorization and Related Topics in the Slice Hyperholomorphic Setting. J. Geom. Anal., 24(2):843–872, 2014.
  12. Schur analysis in the quaternionic setting: the Fueter regular and the slice regular case. In Operator theory. With 51 figures and 2 tables. In 2 volumes, pages 1745–1786. Basel: Springer, 2015.
  13. Slice hyperholomorphic Schur analysis, volume 256 of Operator Theory: Advances and Applications. Basel: Birkhäuser/Springer, 2016.
  14. D. Alpay and P. Dewilde. Time-varying signal approximation and estimation. In M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran, editors, Signal processing, scattering and operator theory, and numerical methods (Amsterdam, 1989), volume 5 of Progress in systems and control theory, pages 1–22. Birkhäuser Boston, Boston, MA, 1990.
  15. Lossless inverse scattering and reproducing kernels for upper triangular operators. In I. Gohberg, editor, Extension and interpolation of linear operators and matrix functions, volume 47 of Oper. Theory Adv. Appl., pages 61–135. Birkhäuser, Basel, 1990.
  16. Nonstationary analogs of the Herglotz representation theorem: the discrete case. J. Funct. Anal., 166(1):85–129, 1999.
  17. Nonstationary analogs of the Herglotz representation theorem: realizations centered at an arbitrary point. In Recent advances in operator theory (Groningen, 1998), volume 124 of Oper. Theory Adv. Appl., pages 63–87. Birkhäuser, Basel, 2001.
  18. Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, volume 96 of Operator theory: Advances and Applications. Birkhäuser Verlag, Basel, 1997.
  19. D. Alpay and H. Dym. On applications of reproducing kernel spaces to the Schur algorithm and rational J𝐽Jitalic_J-unitary factorization. In I. Gohberg, editor, I. Schur methods in operator theory and signal processing, volume 18 of Operator Theory: Advances and Applications, pages 89–159. Birkhäuser Verlag, Basel, 1986.
  20. D. Alpay and I. Gohberg. Unitary rational matrix functions. In I. Gohberg, editor, Topics in interpolation theory of rational matrix-valued functions, volume 33 of Operator Theory: Advances and Applications, pages 175–222. Birkhäuser Verlag, Basel, 1988.
  21. Gleason problem, rational functions and spaces of regular functions: The split-quaternion case. Israel Journal of Mathematics, 226:319–349, 2018.
  22. D. Alpay and Y. Peretz. Realizations for Schur upper triangular operators. In A. Dijksma, I. Gohberg, M.A. Kaashoek, and R. Mennicken, editors, Contributions to operator theory in spaces with an indefinite metric, volume 106 of Operator Theory: Advances and Applications, pages 37–90. Birkhäuser Verlag, Basel, 1998.
  23. D. Alpay and Y. Peretz. Realizations for Schur upper triangular operators centered at an arbitrary point. Integral Equations Operator Theory, 37(3):251–323, 2000.
  24. Reproducing kernel spaces of series of Fueter polynomials. In Operator theory in Krein spaces and nonlinear eigenvalue problems, volume 162 of Oper. Theory Adv. Appl., pages 19–45. Birkhäuser, Basel, 2006.
  25. Carathéodory functions in the Banach space setting. Linear Algebra Appl., 425:700–713, 2007.
  26. N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337–404, 1950.
  27. Interpolation of rational matrix functions, volume 45 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1990.
  28. V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math., 14:187–214, 1961.
  29. Factorization of matrix and operator functions: the state space method, volume 178 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2008. Linear Operators and Linear Systems.
  30. S. Bochner and K. Chandrasekharan. Fourier transforms. Princeton University Press, 1949. Reprinted with permission of the original publishers, Kraus Reprint Corporation, New-York, 1965.
  31. L. de Branges. Complementation in Kreĭn spaces. Trans. Amer. Math. Soc., 305:277–291, 1988.
  32. Canonical models in quantum scattering theory. In C. Wilcox, editor, Perturbation theory and its applications in quantum mechanics, pages 295–392. Wiley, New York, 1966.
  33. Square summable power series. Holt, Rinehart and Winston, New York, 1966.
  34. M. S. Brodskiĭ. Triangular and Jordan representations of linear operators. American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin, Translations of Mathematical Monographs, Vol. 32.
  35. Noncommutative functional calculus, volume 289 of Progress in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2011. Theory and applications of slice hyperholomorphic functions.
  36. L. de Branges. Hilbert spaces of entire functions. Prentice-Hall Inc., Englewood Cliffs, N.J., 1968.
  37. E. Deprettere and P. Dewilde. The generalized Schur algorithm, volume 29 of Operator Theory: Advances and Applications, pages 97–115. Birkhäuser Verlag, Basel, 1988.
  38. P. Dewilde and H. Dym. Interpolation for upper triangular operators. In I. Gohberg, editor, Time–variant systems and interpolation, volume 56 of Oper. Theory Adv. Appl., pages 153–260. Birkhäuser Verlag, Basel, 1992.
  39. W.F. Donoghue. Monotone matrix functions and analytic continuation, volume 207 of Die Grundlehren der mathematischen Wissennschaften. Springer–Verlag, 1974.
  40. H. Dym. J𝐽{J}italic_J–contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989.
  41. On operator ranges. Advances in Mathematics, 7:254–281, 1971.
  42. M. Fliess. Matrices de Hankel. J. Math. Pures Appl. (9), 53:197–222, 1974.
  43. I. Frenkel and M. Libine. Split quaternionic analysis and separation of the series for S⁢L⁢(2,ℝ)𝑆𝐿2ℝSL(2,\mathbb{R})italic_S italic_L ( 2 , blackboard_R ) and S⁢L⁢(2,ℂ)/S⁢L⁢(2,ℝ)𝑆𝐿2ℂ𝑆𝐿2ℝSL(2,\mathbb{C})/SL(2,\mathbb{R})italic_S italic_L ( 2 , blackboard_C ) / italic_S italic_L ( 2 , blackboard_R ). Adv. Math., 228(2):678–763, 2011.
  44. E. Fricain and J. Mashreghi. The theory of ℋℋ\mathcal{H}caligraphic_H(b𝑏bitalic_b) spaces. Vol. 1, volume 20 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2016.
  45. E. Fricain and J. Mashreghi. The theory of ℋ⁢(b)ℋ𝑏\mathcal{H}(b)caligraphic_H ( italic_b ) spaces. Vol. 2, volume 21 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2016.
  46. B. Fritzsche and B. Kirstein, editors. Ausgewählte Arbeiten zu den Ursprüngen der Schur–Analysis, volume 16 of Teubner–Archiv zur Mathematik. B.G. Teubner Verlagsgesellschaft, Stuttgart–Leipzig, 1991.
  47. R. Godement. Les fonctions de type positif et la théorie des groupes. Trans. Amer. Math. Soc., 63:1–84, 1948.
  48. R. Godement. Cours d’algèbre. Hermann, 1987.
  49. Invariant subspaces of matrices with applications. Can. Math. Soc. Ser. Monogr. Adv. Texts. John Wiley, New York, NY, 1986.
  50. J. Górniak and A. Weron. Aronszajn-Kolmogorov type theorems for positive definite kernels in locally convex spaces. Studia Math., 69(3):235–246, 1980/81.
  51. G. Herglotz. Über Potenzenreihen mit positiven reelle Teil im Einheitskreis. Sitzungsber Sachs. Akad. Wiss. Leipzig, Math, 63:501–511, 1911.
  52. M. Libine. An invitation to split quaternionic analysis. In Hypercomplex analysis and applications, Trends Math., pages 161–180. Birkhäuser/Springer Basel AG, Basel, 2011.
  53. Ø. Ore. Theory of non-commutative polynomials. Ann. Math. (2), 34:480–508, 1933.
  54. L. Rodman. An introduction to operator polynomials, volume 38 of Oper. Theory: Adv. Appl. Basel, Birkhäuser Verlag, 1989.
  55. D. Sarason. Sub–Hardy Hilbert spaces in the unit disk, volume 10 of University of Arkansas lecture notes in the mathematical sciences. Wiley, New York, 1994.
  56. F. Sommen. A product and an exponential function in hypercomplex function theory. Applicable Anal., 12(1):13–26, 1981.
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Authors (2)

  1. Daniel Alpay
  2. Ilwoo Cho
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