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Extensions of symmetric operators I: The inner characteristic function case

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Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θB by constructing a bijection between the quotient of this set by a certain natural equivalence relation and the set of all contractive analytic functions φ which are greater or equal to θB.
Opis fizyczny
  • University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
  • [1] N.I. Akhiezer and I.M. Glazman. Theory of Linear Operators in Hilbert Space. Dover Publications, New York, NY, 1993.
  • [2] R.T.W. Martin. Unitary perturbations of compressed n-dimensional shifts. Complex Anal. Oper. Theory, 7:767–799, 2013.
  • [3] R.T.W. Martin. Representation of symmetric operators with deficiency indices (1, 1) in de Branges space. Complex Anal. Oper. Theory, 5:545–577, 2011.
  • [4] A. Aleman, R.T.W. Martin, and W.T. Ross. On a theorem of Livsic. J. Funct. Anal., 264:999–1048, 2013.
  • [5] S.R. Garcia, R.T.W. Martin, and W.T. Ross. Partial orders on partial isometries. Submitted to:J. Oper. Theory, 2015. arXiv:FA/1403.4450
  • [6] M.A. Naimark. Linear differential operators Vol. II. Frederick Ungar Publishing Co., New York, 1969.
  • [7] W.N. Everitt and A. Zettl. Differential operators generated by a countable number of quasi-differential expressions on the real line. Proc. London Math. Soc. 3:524–544, 1992.
  • [8] J. Eckhardt, F. Gesztesy, R. Nichols and G. Teschl. Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials. Opus. Math., 33:467–563, 2013.
  • [9] A. Wang and A. Zettl. Self-adjoint Sturm-Liouville problems with discontinuous boundary conditions. Methods Appl. Anal., 22:1–28, 2015.
  • [10] L.O. Silva, G. Teschl and J. Toloza. Singular Schrödinger operators as self-adjoint extensions of N-entire operators. Proc. Amer. Math. Soc., 143:2103–2115, 2014.
  • [11] P.E.T. Jorgensen and E.P.J. Pearse. Operator theory and analysis of infinite networks. Universitext, Springer, 2008.
  • [12] P.E.T. Jorgensen and E.P.J. Pearse. Resistance boundaries of infinite networks. In: Random Walks, Boundaries and Spectra, 111–142, Springer, 2011.
  • [13] P.E.T. Jorgensen and E.P.J. Pearse. Spectral reciprocity and matrix representations of unbounded operators. J. Funct. Anal., 261:749–776, 2011.
  • [14] P.E.T. Jorgensen and F. Tian. Frames and factorization of graph Laplacians. Opus. Math., 35:293–332, 2015.
  • [15] P.E.T. Jorgensen. Unbounded graph-Laplacians in energy space, and their extensions. J. Appl. Math. Comp., 39:155–187, 2012.
  • [16] P.E.T. Jorgensen and E.P.J. Pearse A Hilbert space approach to effective resistance metric. Complex Anal. Oper. Theory, 4:975–1013, 2010.
  • [17] R. Carlson. Boundary value problems for infinite network graphs. In: Analysis on graphs and its applications Proc. Sympos. Pure Math., 77:355–368, 2015.
  • [18] R. Carlson. Harmonic analysis for graph refinements and the continuous graph FFT. Linear Algebra Appl., 430:2589–2876, 2009.
  • [19] S. Golénia and C. Schumacher. The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs. J. Math. Phys., 52:063512, 2011.
  • [20] P. Exner. A duality between Schrˇodinger operators on graphs and certain Jacobi matrices. In: Annales de l’Institut Henri Poincare-A Physique Theorique, 66:359–372, 1997.
  • [21] L.O. Silva and J. H. Toloza. Applications of Krein’s theory of regular symmetric operators to sampling theory. J. Phys. A, 40:9413, 2007.
  • [22] P.R, Chernoff. Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal., 12:401–414, 1973.
  • [23] H.O. Cordes. Self-adjointness of powers of elliptic operators on non-compact manifolds. Math. Ann., 195, 257–272, 1971.
  • [24] R.T.W. Martin and A. Kempf. Approximation of bandlimited functions on a non-compact manifold by bandlimited functions on compact submanifolds. Samp. Th. Sig. Img. Proc., 7:282–292, 2008.
  • [25] E.P.J. Pearse. Self-similar fractals as boundaries of networks. arXiv:1104.1650, 2011.
  • [26] K. Lau and X. Wang. Self-similar sets as hyperbolic boundaries. Indiana Univ. Math. J., 58:1777–1795, 2009.
  • [27] R.T. Powers. Resistance inequalities for the isotropic Heisenberg ferromagnet. J. Mathematical Phys., 17:1910–1918, 1976.
  • [28] R. T. Powers. Resistance inequalities for KMS states of the isotropic Heisenberg model. Comm. Math. Phys., 51:151–156, 1976.
  • [29] M.L. Gorbachuk and V.I. Gorbachuk, editors. M.G. Krein’s Lectures on Entire Operators. Birkhauser, Boston, 1997.
  • [30] P.R. Halmos and J.E. McLaughlin. Partial isometries. Pacific J. Math., 13:361–371, 1963.
  • [31] B. Sz.-Nagy and C. Foias. Harmonic analysis of operators on Hilbert space. American Elsevier publishing company, Inc., New York, N.Y., 1970.
  • [32] M.S. Livšic. Isometric operators with equal deficiency indices. AMS trans., 13:85–103, 1960.
  • [33] M.S. Livšic. A class of linear operators in Hilbert space. AMS trans., 13:61–83, 1960.
  • [34] L. O. Silva and J. H. Toloza. On the spectral characterization of entire operators with deficiency indices (1,1). J. Math. Anal. Appl., 367:360–373, 2010.
  • [35] U. Habock. Reproducing kernel spaces of entire functions. Diploma Thesis, Technishcen Universitat Wien, 2001.
  • [36] L. de Branges. Hilbert spaces of entire functions. Prentice-Hall, Englewood Cliffs, NJ, 1968.
  • [37] L. de Branges. Perturbations of self-adjoint transformations. Amer. J. Math., 84:543–560, 1962.
  • [38] D. Sarason. Sub-Hardy Hilbert spaces in the unit disk. John Wiley & Sons Inc., New York, NY, 1994.
  • [39] R.T.W.Martin. Semigroups of partial isometries and symmetric operators. Integral Equations Operator Theory, 70:205–226, 2011.
  • [40] V. Paulsen. Completely Bounded Maps and Operator Algebras. Cambridge University Press, New York, NY, 2002.
  • [41] A.B. Aleksandrov. Isometric embeddings of coinvariant subspaces of the shift operator. J. Math. Sci., 92:3543–3549, 1998.
  • [42] V. Paulsen. An Introduction to the theory of reproducing kernel Hilbert spaces. vern/rkhs.pdf, 2009.
  • [43] B. Schroeder. Ordered sets: an introduction. Springer, New York, 2003.
  • [44] S.R. Garcia, R.T.W. Martin, and W.T. Ross. Partial orders on partial isometries. In preparation., 2014.
  • [45] R.T.W. Martin. Near invariance and symmetric operators. Oper. Matrices, 8:513–528, 2014.
  • [46] R.T.W. Martin. Bandlimited functions, curved manifolds and self-adjoint extensions of symmetric operators. Ph.D Thesis, University of Waterloo, 2008.
  • [47] M.A. Naimark. Extremal spectral functions of a symmetric operator. Izvest. Akad. Nauk SSSR, Ser. Mat., 1942.
  • [48] R.C. Gilbert. Extremal spectral functions of a symmetric operator. Pacific J. Math., 14:75–84, 1964.
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