Abstract
In a previous paper, 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 inclusion operator form a symmetric pair.
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References
Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950). 14
Bell, D.: The Malliavin calculus and hypoelliptic differential operators. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18, 1550001, 24 (2015). 2, 7
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1. Springer, New York (1979). 2, 8, 11, 12, 13
Dunford, N., Schwartz, J.T.: Linear Operators. Part II. Wiley Classics Library. Wiley, New York (1988). 2, 3, 12
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
Gawarecki, L.: Transformations of index set for Skorokhod integral with respect to Gaussian processes. J. Appl. Math. Stochastic Anal. 12, 105–111 (1999). 2, 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
Houdayer, C., Vaes, S.: Type III factors with unique Cartan decomposition. J. Math. Pures Appl. 100(9), 564–590 (2013). 2, 11, 12
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
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
Jorgensen, P.E.T., Pearse, E.P.J.: A hilbert space approach to effective resistance metrics. Complex Anal. Oper. Theory 4, 975–1030 (2010). arXiv:0906.2535. 13, 15
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)
Jorgensen, P.E.T., Pearse, E.P.J.: Gel’fand triples and boundaries of infinite networks. New York J. Math. 17, 745–781 (2011). arXiv:0906.2745. 13, 15
Jorgensen, P.E.T., Pearse, E.P.J.: Spectral reciprocity and matrix representations of unbounded operators. J. Funct. Anal. 261, 749–776 (2011). arXiv:0911.0185. 13, 15, 16
Jorgensen, P.E.T., Pearse, E.P.J.: A discrete gauss-green identity for unbounded laplace operators, and the transience of random walks. Israel J. Math. 196, 113–160 (2013). arXiv:0906.1586. 13, 14, 15
Jorgensen, P.E.T., Pearse, E.P.J.: Multiplication operators on the energy space. J. Operator Theory 69, 135–159 (2013). arXiv:1007.3516. 13, 15, 16
Jorgensen, P.E.T., Pearse, E.P.J.: Spectral comparisons between networks with different conductance functions. Journal of Operator Theory 72, 71–86 (2014). arXiv:1107.2786. 13, 15, 16
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
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
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
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)
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)
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
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition. 14
Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. Preprint (2009). arXiv:0904.2985. 13
Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5, 198–224 (2010). 13
Lyons, R., Peres, Y.: Probability on trees and graphs. Unpublished. 13, 15
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
Soardi, P.M.: Potential Theory on Infinite Networks, Volume 1590 of Lecture Notes in Mathematics. Springer, Berlin (1994). 15
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
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
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)
Yamasaki, M.: Discrete potentials on an infinite network. Mem. Fac. Sci. Shimane Univ. 13, 31–44 (1979). 15
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Jorgensen, P.E.T., Pearse, E.P.J. Symmetric Pairs of Unbounded Operators in Hilbert Space, and Their Applications in Mathematical Physics. Math Phys Anal Geom 20, 14 (2017). https://doi.org/10.1007/s11040-017-9245-1
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DOI: https://doi.org/10.1007/s11040-017-9245-1
Keywords
- Graph energy
- Graph Laplacian
- Spectral graph theory
- Resistance network
- Effective resistance
- Hilbert space
- Reproducing kernel
- Unbounded linear operator
- Self-adjoint extension
- Friedrichs extension
- Krein extension
- Essentially self-adjoint
- Spectral resolution
- Defect indices
- Symmetric pair
- Gaussian fields
- Abstract Wiener space
- Malliavin calculus
- Malliavin derivative
- Stochastic integration
- Tomita-Takesaki theory
- Von Neumann algebra
- Type III factor
- Modular automorphism