Abstract
We study the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs. For a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum. We then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs.
Similar content being viewed by others
References
Bär, C., Pacelli Bessa, G.: Stochastic completeness and volume growth. Proc. Am. Math. Soc. 138(7), 2629–2640 (2010). doi:10.1090/S0002-9939-10-10281-0
Barroso, C.S., Pacelli Bessa, G.: Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds. Int. J. Appl. Math. Stat. 6(D06), 82–86 (2006)
Bauer, F., Jost, J., Liu, S.: Ollivier–Ricci curvature and the spectrum of the normalized graph Laplace operator (2012, preprint). arXiv:1105.3803v1[math.CO]
Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discret. Comput. Geom. 25(1), 141–159 (2001)
Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006). doi:10.1515/ADVGEOM.2006.014
Beurling, A., Denny, J.: Espaces de Dirichlet. I. Le cas élémentaire. Acta Math. 99, 203–224 (1958) (French)
Biggs, N.L., Mohar, B., Shawe-Taylor, J.: The spectral radius of infinite graphs. Bull. London Math. Soc. 20(2), 116–120 (1988). doi:10.1112/blms/20.2.116
Bonciocat, A.-I., Sturm, K.-T.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), 2944–2966 (2009). doi:10.1016/j.jfa.2009.01.029
Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178(4), 501–508 (1981). doi:10.1007/BF01174771
Brooks, R.: The spectral geometry of k-regular graphs. J. Anal. Math. 57, 120–151 (1991)
Chavel, I., Karp, L.: Large time behavior of the heat kernel: the parabolic \(\lambda \)-potential alternative. Comment. Math. Helv. 66(4), 541–556 (1991). doi: 10.1007/BF02566664
Cheeger, J., Yau, S.T.: A lower bound for the heat kernel. Commun. Pure Appl. Math. 34(4), 465–480 (1981). doi:10.1002/cpa.3160340404
Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(3), 289–297 (1975)
Dodziuk, J.: Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32(5), 703–716 (1983). doi:10.1512/iumj.1983.32.32046
Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984). doi:10.2307/1999107
Dodziuk, J.: Elliptic operators on infinite graphs. Analysis, geometry and topology of elliptic operators. World Sci. Publ., Hackensack NJ pp. 353–368 (2006)
Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math., vol. 73. Amer. Math. Soc., Providence, pp. 25–40 (1988)
Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, pp. 68–74 (1986)
Dodziuk, J., Mathai, V.: Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel, Contemp. Math., vol. 398, Amer. Math. Soc., Providence RI, pp. 69–81 (2006)
Feller, W.: On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 65(2), 527–570 (1957)
Forman, R.: Bochner’s method for cell complexes and combinatorial Ricci curvature. Discret. Comput. Geom. 29(3), 323–374 (2003). doi:10.1007/s00454-002-0743-x
Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. (2) 48(2), 293–302, (1996). doi:10.2748/tmj/1178225382
Fujiwara, K.: The Laplacian on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996). doi:10.1215/S0012-7094-96-08308-8
Fukushima, M., Ōshima, Y., Masayoshi Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter& Co., Berlin (1994)
Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36(2), 135–249 (1999). doi:10.1090/S0273-0979-99-00776-4
Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes, Mathematische Zeitschrift, 1–29 (2011). doi:10.1007/s00209-011-0911-x
Haeseler, S., Keller, M.: Generalized solutions and spectrum for Dirichlet forms on graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds.) Random Walks, Boundaries and Spectra. Progress in Probability, vol. 64, Birkhäuser Verlag, Basel, pp. 181–199 (2011)
Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2(4), 397–432 (2012)
Harmer, M.: Discreteness of the spectrum of the Laplacian and stochastic incompleteness. J. Geom. Anal. 19(2), 358–372 (2009). doi:10.1007/s12220-008-9055-6
Hasminskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor. Verojatnost. i Primenen. 5, 196–214 (1960) (Russian, with English summary)
Higuchi, Y.: Combinatorial curvature for planar graphs. J. Graph Theory 38(4), 220–229 (2001). doi:10.1002/jgt.10004
Higuchi, Y.: Boundary area growth and the spectrum of discrete Laplacian. Ann. Global Anal. Geom. 24(3), 201–230 (2003). doi:10.1023/A:1024733021533
Huang, X.: A note on Stochastic incompleteness for graphs and weak Omori–Yau maximum principle. J. Math. Anal. Appl. 379(2), 764–782 (2011). doi:10.1016/j.jmaa.2011.02.009
Ichihara, K.: Curvature, geodesics and the Brownian motion on a Riemannian manifold. II. Explosion properties. Nagoya Math. J. 87, 115–125 (1982)
Jost, J., Liu, S.: Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs (2012, preprint). arXiv:1103.4037v2[math.CO]
Keller, M.: The essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51–66 (2010). doi:10.1007/s00208-009-0384-y
Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012). doi:10.1515/CRELLE.2011.122
Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5(4), 198–224 (2010). doi:10.1051/mmnp/20105409
Keller, M., Lenz, D., Vogt, H., Wojciechowski, R.: Note on basic features of large time behaviour of heat kernels (2011, preprint). arXiv:1101. 0373v1[math.FA]
Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. 268, 871–886 (2011). doi:10.1007/s00209-010-0699-0
Li, P.: Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. Math. (2) 124(1), 1–21 (1986). doi:10.2307/1971385
Lin, Y., Yau, S.-T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett. 17(2), 343–356 (2010)
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009). doi:10.4007/annals.2009.169.903
Mohar, B.: Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl. 103, 119–131 (1988). doi:10.1016/0024-3795(88)90224-8
Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009). doi:10.1016/j.jfa.2008.11.001
Pinchover, Y.: Large time behavior of the heat kernel. J. Funct. Anal. 206(1), 191–209 (2004). doi:10.1016/S0022-1236(03)00110-1
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)
Reuter, G.E.H.: Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46 (1957)
Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006). doi:10.1007/s11511-006-0002-8
Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006). doi:10.1007/s11511-006-0003-7
Urakawa, H.: Heat kernel and Green kernel comparison theorems for infinite graphs. J. Funct. Anal. 146(1), 206–235 (1997). doi:10.1006/jfan.1996.3030
Urakawa, H.: Eigenvalue comparison theorems of the discrete Laplacians for a graph. Geom. Dedicata 74(1), 95–112 (1999). doi:10.1023/A:1005008324245
Urakawa, H.: The spectrum of an infinite graph. Can. J. Math. 52(5), 1057–1084 (2000)
Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. J. Math. Anal. Appl. 370(1), 146–158 (2010). doi:10.1016/j.jmaa.2010.04.044
Wojciechowski, R.K.: Stochastic completeness of graphs. ProQuest LLC, Ann Arbor. Ph.D., MI Thesis, City University of New York (2008)
Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), 1419–1441 (2009). doi:10.1512/iumj.2009.58.3575. MR 2542093
Wojciechowski, R.K.: Stochastically incomplete manifolds and graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds.) Random Walks, Boundaries and Spectra, Progress in Probability, vol. 64. Birkhäuser Verlag, Basel, pp. 163–179 (2011)
Żuk, A.: A generalized Følner condition and the norms of random walk operators on groups. Enseign. Math. (2) 45(3–4), 321–348 (1999)
Acknowledgments
The authors are grateful to Józef Dodziuk for his continued support. MK and RW would like to thank the Group of Mathematical Physics of the University of Lisbon for their generous backing while parts of this work were completed. In particular, RW extends his gratitude to Pedro Freitas and Jean-Claude Zambrini for their encouragement and assistance. RW gratefully acknowledges financial support of the FCT in the forms of grant SFRH/BPD/45419/2008 and project PTDC/MAT/101007/2008.
Author information
Authors and Affiliations
Corresponding author
Appendix A: Reducing subspaces and commuting operators
Appendix A: Reducing subspaces and commuting operators
We study symmetries of selfadjoint operators. These symmetries are given in terms of bounded operators commuting with the selfadjoint operator in question. We present a general characterization in Theorem A.1. With Lemma A.5, we then turn to the question of how symmetries of a symmetric non-negative operator carry over to its Friedrichs extension. Finally, we specialize to the situation in which the bounded operator is a projection onto a closed subspace. The main result of this appendix, Corollary A.8, characterizes when a selfadjoint operator commutes with such a projection.
While these results are certainly known in one form or another, we have not found all of them in the literature in the form discussed below. In the main body of the paper they will be applied to Laplacians on graphs. However, they are general enough to be applied to Laplace-Beltrami operators on manifolds as well.
A subspace \(U\) of a Hilbert space is said to be invariant under the bounded operator \(A\) if \(A\) maps \(U\) into \(U\).
Theorem A.1
Let \(L\) be a selfadjoint non-negative operator on the Hilbert space \(\mathcal H \) and \(A\) a bounded operator on \(\mathcal H \). Then, the following assertions are equivalent:
-
(i)
\(D(L)\) is invariant under \(A\) and \(L A x= A Lx \) for all \(x\in D(L)\).
-
(ii)
\(D(L^{1/2})\) is invariant under \(A \) and \(L^{1/2} A x= A L^{1/2}x\) for all \(x\in D(L^{1/2})\).
-
(iii)
\(1_{[0,t]} (L) A = A 1_{[0,t]} (L)\) for all \(t\ge 0\).
-
(iv)
\(e^{- t L} A = A e^{- t L} \) for all \(t\ge 0\).
-
(v)
\( (L + \alpha )^{-1} A = A (L+ \alpha )^{-1}\) for all \(\alpha >0\).
-
(vi)
\( g(L) A = A g(L)\) for all bounded measurable \(g : [0,\infty ) \longrightarrow \mathbb{C }\).
Proof
This is essentially standard. We sketch a proof for the convenience of the reader. We first show that (iii), (iv), (v) and (vi) are all equivalent:
(iii) \(\Longrightarrow \) (iv): This follows by a simple approximation argument.
(iv) \(\Longrightarrow \) (v): This follows immediately from \((L + \alpha )^{-1} = \int _0^\infty e^{- t \alpha } e^{- t L } dt\) (which, in turn, is a direct consequence of the spectral calculus).
(v) \(\Longrightarrow \) (vi): The assumption (v) together with a Stone/Weierstrass-type argument shows that \( g(L) A = A g(L)\) for all continuous \(g : [0,\infty )\longrightarrow \mathbb{C }\) with \(g(x) \rightarrow 0\) for \(x\rightarrow \infty \). Now, it is not hard to see that the set
is closed under pointwise convergence of uniformly bounded sequences. This gives the desired statement (vi).
(vi) \(\Longrightarrow \) (iii): This is obvious.
We now show (ii) \(\Longrightarrow \) (i) \(\Longrightarrow \) (v) and (vi) \(\Longrightarrow \) (ii).
(ii) \(\Longrightarrow \) (i): This is clear as \(L = L^{1/2} L^{1/2}\).
(i) \(\Longrightarrow \) (v): Obviously, (i) implies \(A (L + \alpha ) x = (L + \alpha ) A x \) for all \(\alpha \in \mathbb{R }\) and \(x\in D(L)\). As \((L+\alpha )\) is injective for \(\alpha >0\) we infer for all such \(\alpha \) that
(vi) \(\Longrightarrow \) (ii): For every natural number \(n\) the operator \(L^{1/2} 1_{[0,n]} (L) = (id^{1/2} 1_{[0,n]}) (L)\) is a bounded operator commuting with \(A\) by (vi). Let \(x\in D(L^{1/2})\) be given and set \(x_n :=1_{[0,n]} (L) x\). Then, \(x_n\) belongs to \(D(L^{1/2})\). Moreover, as \( 1_{[0,n]} (L)\) is a projection, we obtain by (vi) that
In particular, \(A x_n\) belongs to \(D(L^{1/2})\) as well. This gives, by (vi) again, that
As \(x\) belongs to \(D(L^{1/2})\), we infer that \( L^{1/2} 1_{[0,n]} (L) x\) converges to \(L^{1/2} x\). Moreover, \(A x_n\) obviously converges to \(A x\). As \(L^{1/2}\) is closed, we obtain that \(A x\) belongs to \(D(L^{1/2})\) as well and \(L^{1/2} A x = L^{1/2} A x\) holds. \(\square \)
Remark
-
(a)
The method to prove (v) \(\Longrightarrow \) (vi) can be strengthened as follows: Let \(L\) be a selfadjoint operator with spectrum \(\Sigma \). Let \(\mathcal B (\Sigma )\) be the algebra of all bounded measurable functions on \(\Sigma .\) A sequence \((f_n)\) in \(\mathcal B (\Sigma )\) is said to converge to \(f\in \mathcal B (\Sigma )\) in the sense of (♣) if the \((f_n)\) are uniformly bounded and converge pointwise to \(f\). Let \(F\) be a subset of \(\mathcal B \) such that \(f(L) A = A f(L)\) holds for all \(f\in F\). If the smallest subalgebra of \(\mathcal B \) which contains \(F\) and is closed under convergence with respect to (♣) is \(\mathcal B \), then \(g(L) A = A g(L)\) for all \(g\in \mathcal B \).
-
(b)
If \(L\) is an arbitrary selfadjoint operator then the equivalence of (i), (iii) and (vi) is still true and the semigroup in (iv) can be replaced by the unitary group and the resolvents in (v) can be replaced by resolvents for \(\alpha \in \mathbb{C }\setminus \mathbb{R }\) (as can easily be seen using (a) of this remark).
Definition A.2
Let \(L\) be a selfadjoint non-negative operator on a Hilbert space \(\mathcal H \) and \(A \) a bounded operator on \(\mathcal H \). Then, \(A\) is said to commute with \(L\) if one of the equivalent statements of the theorem holds.
Corollary A.3
Let \(L\) be a selfadjoint non-negative operator on a Hilbert space \(\mathcal H \) and \(A \) a bounded operator on \(\mathcal H \). Then, \(A\) commutes with \(L\) if and only if its adjoint \(A^*\) commutes with \(L\).
Proof
Take adjoints in (iii) of the previous theorem. \(\square \)
A simple situation in which the previous theorem can be applied is given next.
Proposition A.4
Let \(L\) be a selfadjoint non-negative operator on the Hilbert space \(\mathcal H \) and let \(A\) be a bounded operator on \(\mathcal H \). Let, for each natural number \(n\), a closed subspace \(\mathcal H _n\) of \(\mathcal H \) be given with \(A \mathcal H _n \subset \mathcal H _n\) and \(\overline{\cup _{n} \mathcal H _n} = \mathcal H \). If, for each \(n\), there exists a selfadjoint non-negative operator \(L_n\) from \(\mathcal H _n\) to \(\mathcal H _n\) with \( A L_n = L_n A\) and
for all natural numbers \(n\), \(x\in \mathcal H _n\) and \(\alpha >0\), then \(A L = LA\) holds. A corresponding statement holds with resolvents replaced by the semigroup.
Proof
By assumption we have \(A (L + \alpha )^{-1}x = (L + \alpha )^{-1} A x\) for all \(x\) from the dense set \(\cup _{n} \mathcal H _n\). By boundedness of the respective operators we infer \(A (L + \alpha )^{-1} = (L + \alpha )^{-1} A\) and the statement follows from the previous theorem. \(\square \)
The previous theorem deals with symmetries of a selfadjoint operator \(L\). Often, the selfadjoint operator arises as the Friedrichs extension of a symmetric operator. We next study how symmetries of a symmetric operator carry over to its Friedrichs extension. Specifically, we consider the following situation:
-
(*)
Let \(\mathcal H \) be a Hilbert space with inner product \(\langle \cdot , \cdot \rangle \). Let \(L_0\) be a symmetric operator on \(\mathcal H \) with domain \(D_0\). Let \(Q_0\) be the associated form, i.e., \(Q_0\) is defined on \(D_0 \times D_0\) via \(Q_0 (u,v) :=\langle L_0 u, v\rangle .\) Assume that \(Q_0\) is non-negative, i.e., \(Q_0 (u,u)\ge 0\) for all \(u\in D_0\). Then, \(Q_0\) is closable. Let \(Q\) be the closure of \(Q_0\), \(D(Q)\) the domain of \(Q\) and \(L\) the Friedrichs extension of \(L_0\), i.e., \(L\) is the selfadjoint operator associated to \(Q\).
Lemma A.5
Assume \((*)\). Let \(A\) be a bounded operator on \(\mathcal H \) with \(D_0\) invariant under \(A\) and \(A^*\), \(A L_0 x = L_0 A x\) and \(A^*L_0 x = L_0 A^*x\) for all \(x\in D_0\). Then, the following assertions are equivalent:
-
(i)
\(D(L)\) is invariant under \(A\) and \(A L x = L A x \) for all \(x\in D(L)\).
-
(ii)
\(D(Q)\) is invariant under \(A\) and \(A^*\) and \(Q (A x, y) = Q (x, A^*y)\) for all \(x,y\in D(Q)\).
-
(iii)
There exists a \(C\ge 0\) with both \(Q_0 (A x, Ax) \le C Q_0 (x,x)\) and \(Q_0 (A^*x, A^*x) \le C Q_0(x,x) \) for all \(x\in D_0\).
Proof
(iii) \(\Longrightarrow \) (ii): By \(A L_0 x = L_0 A x\) for all \(x\in D_0\) we infer that \(Q_0 (Ax, y) = Q_0 (x, A^*y)\) for all \(x,y\in D_0\). As \(Q\) is the closure of \(Q_0\), it now suffices to show that both \((A u_n)\) and \((A^*u_n)\) are a Cauchy sequences with respect to the \(Q\)-norm, whenever \((u_n)\) is a Cauchy sequence with respect to the \(Q\)-norm in \(D_0\). This follows directly from (iii).
(ii) \(\Longrightarrow \) (i): Let \(x \in D(L)\) be given. Then, \(x\) belongs to \(D(Q)\) and, by (ii), \(Ax \) belongs to \(D(Q)\) as well. Thus, we can calculate for all \(y\in D(Q)\)
This implies that \(A x\in D(L)\) and \(L A x = A L x\). Hence, we obtain (i).
(i) \(\Longrightarrow \) (iii): From Theorem A.1 and (i) we infer that \(L^{1/2} A x = A L^{1/2}x\) for all \(x\in D(L^{1/2})\). Now, for \(x\in D_0\) it holds that
By \(A x \in D_0\) for \(x\in D_0\) a direct calculation gives
A similar argument shows \(Q_0 (A^*x, A^*x) \le \Vert A^*\Vert ^2 Q_0 (x,x)\). This finishes the proof. \(\square \)
We now turn to the special situation that \(A\) is the projection onto a closed subspace. In this case, some further strengthening of the above result is possible. We first provide an appropriate definition.
Definition A.6
Let \(\mathcal H \) be a Hilbert space and \(S\) a symmetric operator on \(\mathcal H \) with domain \(D(S)\). A closed subspace \( U\) of \(\mathcal H \) with associated orthogonal projection \(P\) is called a reducing subspace for \(S\) if \(D(S)\) is invariant under \(P\) and \( S P x = P S P x\) for all \(x\in D(S)\).
The previous definition is just a commutation condition in the form discussed above as shown in the next lemma.
Definition A.7
Let \(S\) be a symmetric operator on the Hilbert space \(\mathcal H \) and \(P\) be the orthogonal projection onto a closed subspace \(U\) of \(\mathcal H \). Then, the following assertions are equivalent:
-
(i)
\(U\) is a reducing subspace for \(S\).
-
(ii)
\(D(S)\) is invariant under \(P\) and \(S P x = P S x \) holds for all \(x\in D(S)\).
Proof
The implication (ii) \(\Longrightarrow \) (i) is obvious. It remains to show (i) \(\Longrightarrow \) (ii): We first show \( P S y = 0\) for all \(y\in D(S)\) with \(P y = 0\) (i.e., \(y \perp U\)): Choose \(x\in D(S)\) arbitrarily. Then, as \(Px \in D(S) \subset D(S^*)\) we obtain
As \(D(S)\) is dense, we infer \(P S y=0\). Let now \(x\in D(S)\) be arbitrary. Then, \(x = Px + (1-P) x\) and both \(Px\) and \((1-P) x\) belong to \(D(S)\). Thus, we can calculate
This finishes the proof. \(\square \)
We now come to the main result of the appendix dealing with symmetries of symmetric operators in terms of reducing subspaces.
Corollary A.8
Assume \((*)\). Let \( U \) be a closed subspace of \(\mathcal H \) and \(A\) the orthogonal projection onto \( U \). Assume that \(D_0\) is invariant under \(A\). Then, the following assertions are equivalent:
-
(i)
\(U\) is a reducing subspace for \( L_0\), i.e., \( L_0 A x = A L_0 x\) for all \(x\in D_0\).
-
(ii)
\(Q_0 (A x, A y) = Q_0 (A x, y) = Q_0 (x, Ay) \) for all \(x,y\in D_0\).
-
(iii)
\(D(Q)\) is invariant under \(A\) and \(Q (A x, A y) = Q (Ax, y) = Q (x, Ay) \) for all \(x,y\in D(Q)\).
-
(iv)
\( U \) is a reducing subspace for \(L\).
-
(v)
\(A\) commutes with \( e^{-t L}\) for every \(t\ge 0\).
-
(vi)
\(A\) commutes with \((L+\alpha )^{-1}\) for any \(\alpha >0\).
Proof
Obviously, (i) and (ii) are equivalent. The equivalence of (iii) and (iv) follows from the equivalence of (i) and (ii) in Lemma A.5. The equivalence between (iv), (v) and (vi) follows immediately from Theorem A.1. The implication (iii) \(\Longrightarrow \) (ii) is clear (as \(A D_0 \subseteq D_0\)). It remains to show (ii) \(\Longrightarrow \) (iii): A direct calculation using (ii) gives for all \(x\in D_0\) that
This shows
for all \(x\in D_0\). Now, the implication (iii) \(\Longrightarrow \) (ii) from Lemma A.5 gives (iii). \(\square \)
Rights and permissions
About this article
Cite this article
Keller, M., Lenz, D. & Wojciechowski, R.K. Volume growth, spectrum and stochastic completeness of infinite graphs. Math. Z. 274, 905–932 (2013). https://doi.org/10.1007/s00209-012-1101-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-012-1101-1