Tohoku Mathematical Journal

Interpolation and complex symmetry

Stephan R. Garcia and Mihai Putinar

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Abstract

In a separable complex Hilbert space endowed with an isometric conjugate-linear involution, we study sequences orthonormal with respect to an associated bilinear form. Properties of such sequences are measured by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix. Although developed in an abstract setting, this framework is relevant to a variety of eigenvector interpolation problems arising in function theory and in the study of differential operators.

Article information

Source
Tohoku Math. J. (2), Volume 60, Number 3 (2008), 423-440.

Dates
First available in Project Euclid: 3 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1223057737

Digital Object Identifier
doi:10.2748/tmj/1223057737

Mathematical Reviews number (MathSciNet)
MR2453732

Zentralblatt MATH identifier
1171.30011

Subjects
Primary: 30D55
Secondary: 47A15: Invariant subspaces [See also 47A46]

Keywords
Complex symmetric operator interpolation eigensystem eigenfunction contraction conjugation dissipative operator bilinear form inner function compressed Toeplitz operator

Citation

Garcia, Stephan R.; Putinar, Mihai. Interpolation and complex symmetry. Tohoku Math. J. (2) 60 (2008), no. 3, 423--440. doi:10.2748/tmj/1223057737. https://projecteuclid.org/euclid.tmj/1223057737


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References

  • N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover, New York, 1993.
  • N. Chevrot, E. Fricain and D. Timotin, The characteristic function of a complex symmetric contraction, Proc. Amer. Math. Soc. 135(2007), 2877--2886.
  • O. Christensen, An introduction to frames and Riesz bases, Birkhäuser, Boston, 2003.
  • S. R. Garcia, Conjugation and Clark operators, Contemp. Math. 93(2006), 67--112.
  • S. R. Garcia, The eigenstructure of complex symmetric operators, Proceedings of the 16th International Workshop on Operator Theory and Applications (IWOTA 2005), Birkhäuser Series on Operator Theory (to appear).
  • S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358(2006), 1285--1315.
  • S. R. Garcia and M. Putinar, Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359(2007), 3913--3931.
  • S. R. Garcia and W. R. Wogen, Some new classes of complex symmetric operators, (preprint).
  • J. B. Garnett, Bounded analytic functions (revised first edition), Graduate Texts in Mathematics 236, Springer, New York, 2007.
  • T. M. Gilbreath and W. R. Wogen, Remarks on the structure of complex symmetric operators, Integral Equations Operator Theory 59 (2007), 585--590.
  • I. M. Glazman, On the expansibility in an eigenfunction system of dissipative operators (in Russian), Uspehi Mat. Nauk. 13(1958), 179--181.
  • I. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Amer. Math. Soc., Providence, R.I., 1969.
  • N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, American Mathematical Society, Providence, R.I., 2002.
  • V. V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
  • E. Prodan, S. R. Garcia and M. Putinar, Norm estimates of complex symmetric operators applied to quantum systems, J. Phys. A: Math. Gen. 39(2006), 389--400.
  • M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, selfadjointness, Academic Press, New York, 1975; Part IV: Analysis of operators, Academic Press, New York, 1978.
  • F. Riesz and B. Sz.-Nagy, Functional analysis, Dover, New York, 1990.
  • D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1(2007), 491--526.
  • K. Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series vol. 33, Amer. Math. Soc., Providence, R.I., 2004.