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Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics. (English) Zbl 1424.47055

Summary: In [Complex Anal. Oper. Theory 10, No. 7, 1535–1550 (2016; Zbl 1353.05080)], the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators \(A\) and \(B\), with \(A\subseteq B^\star\) and \(B\subseteq A^\star\). In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We also prove that the basic involutions of Tomita-Takesaki theory are closable and that their closures are mutually adjoint. Applications to functions of finite energy on infinite graphs are also discussed, wherein the Laplace operator and the inclusion operator form a symmetric pair.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
46L36 Classification of factors
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
60H07 Stochastic calculus of variations and the Malliavin calculus
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C63 Infinite graphs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Citations:

Zbl 1353.05080
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References:

[1] Aronszajn, N, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68, 337-404, (1950) · Zbl 0037.20701 · doi:10.1090/S0002-9947-1950-0051437-7
[2] Bell, D, The Malliavin calculus and hypoelliptic differential operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18, 1550001, 24, (2015) · Zbl 1310.60069 · doi:10.1142/S0219025715500010
[3] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1. Springer, New York (1979). 2, 8, 11, 12, 13 · Zbl 0421.46048
[4] Dunford, N., Schwartz, J.T.: Linear Operators. Part II. Wiley Classics Library. Wiley, New York (1988). 2, 3, 12 · Zbl 0635.47002
[5] Fukushima, M., Ōshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Volume 19 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (1994). 14 · Zbl 0838.31001 · doi:10.1515/9783110889741
[6] Gawarecki, L, Transformations of index set for Skorokhod integral with respect to Gaussian processes, J. Appl. Math. Stochastic Anal., 12, 105-111, (1999) · Zbl 0939.60055 · doi:10.1155/S1048953399000118
[7] Hida, T.: Brownian Motion, Volume 11 of Applications of Mathematics. Springer, New York (1980). Translated from the Japanese by the author and T. P. Speed. 2, 7, 8 · Zbl 0432.60002
[8] Houdayer, C; Vaes, S, Type III factors with unique Cartan decomposition, J. Math. Pures Appl., 100, 564-590, (2013) · Zbl 1291.46052 · doi:10.1016/j.matpur.2013.01.013
[9] Jorgensen, P.E.T., Pearse, E.P.J.: Operator theory and analysis of infinite resistance networks, 1-247 (2009). arXiv:0806.3881, 13, 14, 15, 16
[10] Jorgensen, P.E.T., Pearse, E.P.J.: Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian, 1-24 (2009). arXiv:1504.01332, 13, 15
[11] Jorgensen, PET; Pearse, EPJ, A Hilbert space approach to effective resistance metrics, Complex Anal. Oper. Theory, 4, 975-1030, (2010) · Zbl 1209.05144 · doi:10.1007/s11785-009-0041-1
[12] Jorgensen, P.E.T., Pearse, E.P.J.: Resistance boundaries of infinite networks Progress in Probability: Boundaries and Spectral Theory. arXiv:0909.1518. 13, 15, vol. 64, pp 113-143. Birkhauser (2010) · Zbl 0416.31012
[13] Jorgensen, PET; Pearse, EPJ, Gel’fand triples and boundaries of infinite networks, New York J. Math., 17, 745-781, (2011) · Zbl 1241.05080
[14] Jorgensen, PET; Pearse, EPJ, Spectral reciprocity and matrix representations of unbounded operators, J. Funct. Anal., 261, 749-776, (2011) · Zbl 1257.47025 · doi:10.1016/j.jfa.2011.01.016
[15] Jorgensen, PET; Pearse, EPJ, A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks, Israel J. Math., 196, 113-160, (2013) · Zbl 1275.05054 · doi:10.1007/s11856-012-0165-2
[16] Jorgensen, PET; Pearse, EPJ, Multiplication operators on the energy space, J. Operator Theory, 69, 135-159, (2013) · Zbl 1278.47029 · doi:10.7900/jot.2010jul20.1886
[17] Jorgensen, PET; Pearse, EPJ, Spectral comparisons between networks with different conductance functions, Journal of Operator Theory, 72, 71-86, (2014) · Zbl 1389.47069 · doi:10.7900/jot.2012oct05.1978
[18] Jorgensen, P.E.T., Pearse, E.P.J: Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks. to appear Complex Anal. Oper. Theory, 1-11 (2015). arXiv:1512.03463, 1, 2, 3, 14, 16
[19] Jorgensen, P.E.T., Pearse, E.P.J.: Unbounded containment in the energy space of a network and the krein extension of the energy Laplacian. 17 pages, in review (2015). arXiv:1504.01332. 16
[20] Jorgensen, P.E.T., Pearse, E.P.J., Tian, F.: Duality for unbounded operators, and applications. In review, 1-14 (2015). arXiv:1509.08024. 2 · Zbl 1278.47029
[21] Kadison, R.V.: Dual cones and Tomita-Takesaki theory. In Operator algebras and operator theory (Shanghai, 1997), volume 228 of Contemp. Math., pp. 151-178. Amer. Math. Soc., Providence, RI,. 2, 11 (1998) · Zbl 0935.46051
[22] Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras. Vol. II, volume 100 of Pure and Applied Mathematics. Academic Press, Inc., Orlando, FL. Advanced theory. 2, 11 (1986) · Zbl 0601.46054
[23] Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras. Vol. I, volume 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1997). Elementary theory, Reprint of the 1983 original. 3, 11 · Zbl 0888.46039
[24] Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition. 14 · doi:10.1007/978-3-642-66282-9
[25] Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. Preprint (2009). arXiv:0904.2985. 13 · Zbl 1252.47090
[26] Keller, M; Lenz, D, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom., 5, 198-224, (2010) · Zbl 1207.47032 · doi:10.1051/mmnp/20105409
[27] Lyons, R., Peres, Y.: Probability on trees and graphs. Unpublished. 13, 15 · Zbl 1376.05002
[28] Parthasarathy, K.R., Schmidt, K.: Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory Lecture Notes in Mathematics, vol. 272. Springer, Berlin (1972). 2, 8 · Zbl 0237.43005
[29] Soardi, P.M.: Potential Theory on Infinite Networks, Volume 1590 of Lecture Notes in Mathematics. Springer, Berlin (1994). 15
[30] Takesaki, M.: Tomita’s Theory of Modular Hilbert Algebras and its Applications. Lecture Notes in Mathematics, vol. 128. Springer-Verlag, Berlin-New York (1970). 2, 11 · Zbl 0193.42502
[31] Takesaki, M.: Theory of Operator Algebras. II, Volume 125 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2003). Operator Algebras and Non-commutative Geometry, 6. 2, 11 · doi:10.1007/978-3-662-10451-4
[32] van Daele, A.: The Tomita-Takesaki theory for von Neumann algebras with a separating and cyclic vector \(C\)\^{}{∗}-Algebras and Their Applications to Statistical Mechanics and Quantum Field Theory (Proc. Internat. School of Physics “Enrico Fermi”, Course LX, Varenna, 1973). 2, 11, pp 19-28. North-Holland, Amsterdam (1976)
[33] Yamasaki, M, Discrete potentials on an infinite network, Mem. Fac. Sci. Shimane Univ., 13, 31-44, (1979) · Zbl 0416.31012
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