Opuscula Math. 35, no. 3 (2015), 293-332
http://dx.doi.org/10.7494/OpMath.2015.35.3.293

Opuscula Mathematica

# Frames and factorization of graph Laplacians

Palle Jorgensen
Feng Tian

Abstract. Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $$\mathscr{H}_{E}$$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $$\mathscr{H}_{E}$$ we characterize the Friedrichs extension of the $$\mathscr{H}_{E}$$-graph Laplacian. We consider infinite connected network-graphs $$G=\left(V,E\right)$$, $$V$$ for vertices, and $$E$$ for edges. To every conductance function $$c$$ on the edges $$E$$ of $$G$$, there is an associated pair $$\left(\mathscr{H}_{E},\Delta\right)$$ where $$\mathscr{H}_{E}$$ in an energy Hilbert space, and $$\Delta\left(=\Delta_{c}\right)$$ is the $$c$$-Graph Laplacian; both depending on the choice of conductance function $$c$$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $$\mathscr{H}_{E}$$ consisting of dipoles. Now $$\Delta$$ is a well-defined semibounded Hermitian operator in both of the Hilbert $$l^{2}\left(V\right)$$ and $$\mathscr{H}_{E}$$. It is known to automatically be essentially selfadjoint as an $$l^{2}\left(V\right)$$-operator, but generally not as an $$\mathscr{H}_{E}$$ operator. Hence as an $$\mathscr{H}_{E}$$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $$l^{2}\left(V\right)$$.

Keywords: unbounded operators, deficiency-indices, Hilbert space, boundary values, weighted graph, reproducing kernel, Dirichlet form, graph Laplacian, resistance network, harmonic analysis, harmonic analysis, frame, Parseval frame, Friedrichs extension, reversible random walk, resistance distance, energy Hilbert space.

Mathematics Subject Classification: 47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 46N20, 22E70, 31A15, 58J65, 81S25.

Full text (pdf)

1. N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell, Reprint of the 1961 and 1963 translations, Two volumes bound as one.
2. D. Alpay, P. Jorgensen, R. Seager, D. Volok, On discrete analytic functions: products, rational functions and reproducing kernels, J. Appl. Math. Comput. 41 (2013) 1-2, 393-426.
3. V. Anandam, Harmonic functions and potentials on finite or infinite networks, vol. 12 of Lecture Notes of the Unione Matematica Italiana, Springer, Heidelberg; UMI, Bologna, 2011.
4. M.T. Barlow, Random walks, electrical resistance, and nested fractals, [in:] Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), Longman Sci. Tech., Harlow, vol. 283 of Pitman Res. Notes Math. Ser., 131-157, 1993.
5. I. Cho, P.E.T. Jorgensen, Free probability on operator algebras induced by currents in electric resistance networks, Int. J. Funct. Anal. Oper. Theory Appl. 4 (2012) 1, 1-50.
6. I. Cho, P.E.T. Jorgensen, Operators induced by graphs, Lett. Math. Phys. 102 (2012) 3, 323-369.
7. H. Cossette, D. Landriault, É. Marceau, Ruin probabilities in the compound Markov binomial model, Scand. Actuar. J. 4 (2003), 301-323.
8. B. Currey, A. Mayeli, The Orthonormal dilation property for Abstract Parseval wavelet frames, Canad. Math. Bull. 56 (2013) 4, 729-736.
9. P.G. Doyle, J.L. Snell, Random walks and electric networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984.
10. N. Dunford, J.T. Schwartz, Linear Operators. Part II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space, with the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication.
11. M. Durand, Architecture of optimal transport networks, Phys. Rev. E (3), 73 (2006) 1, 016116, 6.
12. M. Ehler, K. A. Okoudjou, Probabilistic Frames: an Overview, [in:] Finite Frames, Birkhäuser/Springer, New York, Appl. Numer. Harmon. Anal., 415-436, 2013.
13. M. Folz, Volume growth and spectrum for general graph Laplacians, Math. Z. 276 (2014) 1-2, 115-131.
14. M.J. Gander, S. Loisel, D.B. Szyld, An optimal block iterative method and preconditioner for banded matrices with applications to PDEs on irregular domains, SIAM J. Matrix Anal. Appl. 33 (2012) 2, 653-680.
15. G. Grimmett, Probability on Graphs, vol. 1 of Institute of Mathematical Statistics Textbooks, Cambridge University Press, Cambridge, 2010.
16. D. Han, W. Jing, D. Larson, P. Li, R.N. Mohapatra, Dilation of dual frame pairs in Hilbert $$C^*$$-modules, Results Math. 63 (2013) 1-2, 241-250.
17. P.E.T. Jorgensen, Essential self-adjointness of the graph-Laplacian, J. Math. Phys. 49 (2008) 7, 073510, 33.
18. P.E.T. Jorgensen, A.M. Paolucci, $$q$$-frames and Bessel functions, Numer. Funct. Anal. Optim. 33 (2012) 7-9, 1063-1069.
19. P.E.T. Jorgensen, E.P.J. Pearse, A Hilbert space approach to effective resistance metric, Complex Anal. Oper. Theory 4 (2010) 4, 975-1013.
20. P.E.T. Jorgensen, E.P.J. Pearse, Resistance Boundaries of Infinite Networks, [in:] Random Walks, Boundaries and Spectra, Birkhäuser, Springer Basel AG, Basel, vol. 64 of Progr. Probab., 111-142, 2011.
21. P.E.T. Jorgensen, E.P.J. Pearse, Spectral reciprocity and matrix representations of unbounded operators, J. Funct. Anal. 261 (2011) 3, 749-776.
22. P.E.T. Jorgensen, E.P.J. Pearse, A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks, Israel J. Math. 196 (2013) 1, 113-160.
23. P.E.T. Jorgensen, M.-S. Song, Comparison of Discrete and Continuous Wavelet Transforms, [in:] Computational Complexity. Vols. 1-6, Springer, New York, 513-526, 2012.
24. V. Kaftal, D.R. Larson, S. Zhang, Operator-valued frames, Trans. Amer. Math. Soc. 361 (2009) 12, 6349-6385.
25. G. Kutyniok, K.A. Okoudjou, F. Philipp, E.K. Tuley, Scalable frames, Linear Algebra Appl. 438 (2013) 5, 2225-2238.
26. I.M. Longini, Jr., A chain binomial model of endemicity, Math. Biosci. 50 (1980) 1-2, 85-93.
27. M. Longla, C. Peligrad, M. Peligrad, On the functional central limit theorem for reversible Markov chains with nonlinear growth of the variance, J. Appl. Probab. 49 (2012) 4, 1091-1105.
28. F.G. Meyer, X. Shen, Perturbation of the eigenvectors of the graph Laplacian: application to image denoising, Appl. Comput. Harmon. Anal. 36 (2014) 2, 326-334.
29. C.S. J.A. Nash-Williams, Random walk and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959), 181-194.
30. F.A. Shah, L. Debnath, Tight wavelet frames on local fields, Analysis (Berlin) 33 (2013) 3, 293-307.
31. M.N.S. Swamy, K. Thulasiraman, Graphs, Networks, and Algorithms, John Wiley & Sons, Inc., New York, 1981.
32. R. Terakado, Construction of constant resistance networks using the properties of two-dimensional regions with antisymmetry, IEEE Trans. Circuits and Systems CAS-25 (1978) 2, 109-111.
33. P. Tetali, Random walks and the effective resistance of networks, J. Theoret. Probab. 4 (1991) 1, 101-109.
34. K.-C. Yuen, J. Guo, Some results on the compound Markov binomial model, Scand. Actuar. J. 3 (2006), 129-140.
35. A.H. Zemanian, Random walks on finitely structured transfinite networks, Potential Anal. 5 (1996) 4, 357-382.
• Palle Jorgensen
• The University of Iowa, Department of Mathematics, Iowa City, IA 52242-1419, USA
• Feng Tian
• Wright State University, Department of Mathematics, Dayton, OH 45435, USA
• Communicated by P.A. Cojuhari.
• Accepted: 2014-11-04.
• Published online: 2014-12-15. 