Abstract
We study unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify additional structures that throw light on the problem. We will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space.
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Alpay D., Levanony D., Rational functions associated with the white noise space and related topics, Potential Anal., 2008, 29,2, 195–220
Alpay D., Bruinsma P., Dijksma A., de Snoo H., Interpolation problems, extensions of symmetric operators and reproducing kernel spaces, I, Oper. Theory Adv. Appl., Vol. 50, Birkhäuser, Basel, 1991
Alpay D., Bruinsma P., Dijksma A., de Snoo H., Addendum: “Interpolation problems, extensions of symmetric operators and reproducing kernel spaces, II”, Integral Equations Operator Theory, 1992, 15(3), 378–388
Alpay D., Levanony D., On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions, Potential Anal., 2008, 28,2, 163–184
Alpay D., Shapiro M., Volok D., Reproducing kernel spaces of series of Fueter polynomials, In Operator theory in Krein spaces and nonlinear eigenvalue problems, Oper. Theory Adv. Appl., Vol. 162, Birkhäuser, Basel, 2006
Aronszajn N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 1950, 68, 337–404
Atkinson K., Han W., Theoretical numerical analysis, Texts in Applied Mathematics, A functional analysis framework, 2nd ed., Vol. 39, Springer, New York, 2005
Baladi V., Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, Vol. 16, World Scientific Publishing Co. Inc., River Edge, NJ, 2000
Barlow M., Bass R., Chen Z-Q, Kassmann M., Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 2009, 361(4), 1963–1999
Behrndt J., Hassi S., de Snoo H., Functional models for Nevanlinna families, Opuscula Math., 2008, 28(3), 233–245
Behrndt J., Hassi S., de Snoo H., Boundary relations, unitary colligations, and functional models, Complex Anal. Oper. Theory, 2009, 3(1), 57–98
Brofferio S., Woess W., Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs, Ann. Inst. H. Poincaré Probab. Statist., 2005, 41(6), 1101–1123
Carlson R., Pivovarchik V., Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A, 2008, 41(14), 145202, 16
Doob J., Discrete potential theory and boundaries, J. Math. Mech., 1959, 8, 433–458
Dunford N., Schwartz J., Linear operators, Part II, John Wiley & Sons Inc., New York, 1988
Hassi S., de Snoo H., Szafraniec F., Componentwise and canonical decompositions of linear relations, Dissertationes Mathematicae, 465, 2009
Hida T., Brownian motion, Volume 11, Applications of Mathematics, Translated from the Japanese by the author and T. P. Speed, Springer-Verlag, New York, 1980
Jorgensen P., Pearse E., Operator theory of electrical resistance networks, preprint available at http://arxiv.org/abs/0806.3881
Klopp F., Pankrashkin K., Localization on quantum graphs with random vertex couplings, J. Stat. Phys., 2008, 131,4, 651–673
Kolmogoroff A., Grundbegriffe der Wahrscheinlichkeitsrechnung, Reprint of the 1933 original, Springer-Verlag, Berlin, 1977
Lax P., Phillips R., Scattering theory for automorphic functions, Bull. Amer. Math. Soc. (N.S.), 1980, 2(2), 261–295
Nelson E., The free Markoff field, J. Functional Analysis, 1973, 12, 211–227
Ortner R., Woess W., Non-backtracking random walks and cogrowth of graphs, Canad. J. Math., 2007, 59(4), 828–844
Reed M., Simon B., Methods of modern mathematical physics, II, Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975
Stone M., Linear transformations in Hilbert space, Vol. 15, American Mathematical Society Colloquium Publications, Reprint of the 1932 original, American Mathematical Society, Providence, RI, 1990
von Neumann J., Über adjungierte Funktionaloperatoren, Ann. of Math. (2), 1932, 33(2), 294–310
Yamasaki K., Nagahama H., Energy integral in fracture mechanics (J-integral) and Gauss-Bonnet theorem, ZAMM Z. Angew. Math. Mech., 2008, 88(6), 515–520
Zhang H., Orthogonality from disjoint support in reproducing kernel Hilbert spaces, J. Math. Anal. Appl., 2009, 349(1), 201–210
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Jorgensen, P.E.T. Unbounded Hermitian operators and relative reproducing kernel Hilbert space. centr.eur.j.math. 8, 569–596 (2010). https://doi.org/10.2478/s11533-010-0021-8
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DOI: https://doi.org/10.2478/s11533-010-0021-8