×

zbMATH — the first resource for mathematics

Unbounded Hermitian operators and relative reproducing kernel Hilbert space. (English) Zbl 1220.47029
The paper is concerned with the theory of selfadjoint extensions of hermitian linear operators on a Hilbert space. The main application is the Laplace operator on a infinite weighted graph.
A duality pair \((\Delta, \mathcal C)\) consists of a densely defined hermitian operator \(\Delta\) on a Hilbert space \(H\) and a closed subspace \(\mathcal C\) with \(\mathcal C \subseteq \ker(\Delta^*)\). In this case, \(\Delta\) admits a hermitian extension \(\Delta_{\mathcal C}\) to \(\mathcal D(\Delta)+\mathcal C\) and the domain of its adjoint is \(\mathcal D(\Delta_{\mathcal C}^*) = H \ominus \mathcal C\).
The author investigates the reproducing kernel Hilbert space with base point \(o\), \((H,X,o)\), under the additional assumptions that all Dirac functions \(\delta_x\), \(x\in X\), belong to \(H\) and that \(H = \overline{\text{span}}\{k(\cdot, x) | x\in X, x\neq o\}\) (\(k\) is the kernel function of \(H\)). He shows that, under this condition, the operator \(\Delta|_V\) defined by \((\Delta|_V f)(x) = \langle \delta_x,\, f\rangle\) with domain \(\text{span}\{ k_x - k_o| x\in X, x\neq o\}\) is densely defined and positive. Moreover, \((\Delta|_V, \mathcal C)\) is a duality pair, where \(\mathcal C = \{ u \in H : \langle y,\, \delta_x \rangle = 0,\;x \in X\}\), so \(\Delta|_V\) can be extended to a hermitian operator \(\Delta_H\) on \(\mathcal D(\Delta|_V) + \mathcal C\). It is shown that \(\Delta_H\) is essentially selfadjoint if and only if \(\Delta|_F\) with domain \(\mathcal D(\Delta|_F) = \text{span}\{ \delta_x | x\in X\}\) is essentially selfadjoint. The paper concludes with an application to an infinite weighted graph.

MSC:
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B25 Linear symmetric and selfadjoint operators (unbounded)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alpay D., Levanony D., Rational functions associated with the white noise space and related topics, Potential Anal., 2008, 29,2, 195-220 http://dx.doi.org/10.1007/s11118-008-9094-4 · Zbl 1203.30051
[2] 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 · Zbl 0737.47016
[3] 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 http://dx.doi.org/10.1007/BF01200325 · Zbl 0780.47016
[4] 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 http://dx.doi.org/10.1007/s11118-007-9070-4 · Zbl 1136.46022
[5] 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
[6] Aronszajn N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 1950, 68, 337-404 http://dx.doi.org/10.2307/1990404 · Zbl 0037.20701
[7] Atkinson K., Han W., Theoretical numerical analysis, Texts in Applied Mathematics, A functional analysis framework, 2nd ed., Vol. 39, Springer, New York, 2005 · Zbl 1068.47091
[8] 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 · Zbl 1012.37015
[9] 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 http://dx.doi.org/10.1090/S0002-9947-08-04544-3 · Zbl 1166.60045
[10] Behrndt J., Hassi S., de Snoo H., Functional models for Nevanlinna families, Opuscula Math., 2008, 28(3), 233-245 · Zbl 1183.47004
[11] Behrndt J., Hassi S., de Snoo H., Boundary relations, unitary colligations, and functional models, Complex Anal. Oper. Theory, 2009, 3(1), 57-98 http://dx.doi.org/10.1007/s11785-008-0064-z · Zbl 1186.47007
[12] 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 http://dx.doi.org/10.1016/j.anihpb.2004.12.004 · Zbl 1083.60062
[13] Carlson R., Pivovarchik V., Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A, 2008, 41(14), 145202, 16 http://dx.doi.org/10.1088/1751-8113/41/14/145202 · Zbl 1149.34021
[14] Doob J., Discrete potential theory and boundaries, J. Math. Mech., 1959, 8, 433-458 · Zbl 0101.11503
[15] Dunford N., Schwartz J., Linear operators, Part II, John Wiley & Sons Inc., New York, 1988 · Zbl 0635.47002
[16] Hassi S., de Snoo H., Szafraniec F., Componentwise and canonical decompositions of linear relations, Dissertationes Mathematicae, 465, 2009 · Zbl 1225.47004
[17] 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 · Zbl 0423.60063
[18] Jorgensen P., Pearse E., Operator theory of electrical resistance networks, preprint available at http://arxiv.org/abs/0806.3881
[19] Klopp F., Pankrashkin K., Localization on quantum graphs with random vertex couplings, J. Stat. Phys., 2008, 131,4, 651-673 http://dx.doi.org/10.1007/s10955-008-9517-z · Zbl 1144.82061
[20] Kolmogoroff A., Grundbegriffe der Wahrscheinlichkeitsrechnung, Reprint of the 1933 original, Springer-Verlag, Berlin, 1977
[21] Lax P., Phillips R., Scattering theory for automorphic functions, Bull. Amer. Math. Soc. (N.S.), 1980, 2(2), 261-295 http://dx.doi.org/10.1090/S0273-0979-1980-14735-7 · Zbl 0442.10018
[22] Nelson E., The free Markoff field, J. Functional Analysis, 1973, 12, 211-227 http://dx.doi.org/10.1016/0022-1236(73)90025-6
[23] Ortner R., Woess W., Non-backtracking random walks and cogrowth of graphs, Canad. J. Math., 2007, 59(4), 828-844 · Zbl 1123.05081
[24] Reed M., Simon B., Methods of modern mathematical physics, II, Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975 · Zbl 0308.47002
[25] 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 · Zbl 0933.47001
[26] von Neumann J., Über adjungierte Funktionaloperatoren, Ann. of Math. (2), 1932, 33(2), 294-310 http://dx.doi.org/10.2307/1968331 · Zbl 0004.21603
[27] 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 http://dx.doi.org/10.1002/zamm.200700140 · Zbl 1138.74004
[28] Zhang H., Orthogonality from disjoint support in reproducing kernel Hilbert spaces, J. Math. Anal. Appl., 2009, 349(1), 201-210 http://dx.doi.org/10.1016/j.jmaa.2008.08.030 · Zbl 1159.46014
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.