# zbMATH — the first resource for mathematics

Interpolation and complex symmetry. (English) Zbl 1171.30011
Summary: 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.

##### MSC:
 30D55 $$H^p$$-classes (MSC2000) 47A15 Invariant subspaces of linear operators
Full Text:
##### References:
 [1] N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover, New York, 1993. · Zbl 0874.47001 [2] N. Chevrot, E. Fricain and D. Timotin, The characteristic function of a complex symmetric contraction, Proc. Amer. Math. Soc. 135(2007), 2877–2886. · Zbl 1126.47010 [3] O. Christensen, An introduction to frames and Riesz bases, Birkhäuser, Boston, 2003. · Zbl 1017.42022 [4] S. R. Garcia, Conjugation and Clark operators, Contemp. Math. 93(2006), 67–112. · Zbl 1099.30020 [5] 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). · Zbl 1168.47021 [6] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358(2006), 1285–1315. · Zbl 1087.30031 [7] S. R. Garcia and M. Putinar, Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359(2007), 3913–3931. · Zbl 1123.47030 [8] S. R. Garcia and W. R. Wogen, Some new classes of complex symmetric operators, (preprint). · Zbl 1208.47036 [9] J. B. Garnett, Bounded analytic functions (revised first edition), Graduate Texts in Mathematics 236, Springer, New York, 2007. [10] T. M. Gilbreath and W. R. Wogen, Remarks on the structure of complex symmetric operators, Integral Equations Operator Theory 59 (2007), 585–590. · Zbl 1135.47032 [11] I. M. Glazman, On the expansibility in an eigenfunction system of dissipative operators (in Russian), Uspehi Mat. Nauk. 13(1958), 179–181. · Zbl 0081.12204 [12] I. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Amer. Math. Soc., Providence, R.I., 1969. · Zbl 0181.13504 [13] 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. [14] V. V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. · Zbl 1030.47002 [15] 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. · Zbl 1088.81045 [16] 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. · Zbl 0308.47002 [17] F. Riesz and B. Sz.-Nagy, Functional analysis, Dover, New York, 1990. · Zbl 0732.47001 [18] D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1(2007), 491–526. · Zbl 1144.47026 [19] K. Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series vol. 33, Amer. Math. Soc., Providence, R.I., 2004. · Zbl 1057.30036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.