Spectral gap for hyperbounded operators
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Abstract:
Let $(E,\mathcal F,\mu )$ be a probability space, and $P$ a symmetric linear contraction operator on $L^2(\mu )$ with $P1=1$ and $\|P\|_{L^2(\mu )\to L^4(\mu )}<\infty$. We prove that $\|P\|_{L^2(\mu )\to L^4(\mu )}^4<2$ is the optimal sufficient condition for $P$ to have a spectral gap. Moreover, the optimal sufficient conditions are obtained, respectively, for the defective log-Sobolev and for the defective Poincaré inequality to imply the existence of a spectral gap. Finally, we construct a symmetric, hyperbounded, ergodic contraction $C_0$-semigroup without a spectral gap.References
- Shigeki Aida, Uniform positivity improving property, Sobolev inequalities, and spectral gaps, J. Funct. Anal. 158 (1998), no. 1, 152–185. MR 1641566, DOI 10.1006/jfan.1998.3286
- D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New trends in stochastic analysis (Charingworth, 1994) World Sci. Publ., River Edge, NJ, 1997, pp. 43–75. MR 1654503
- E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), no. 2, 335–395. MR 766493, DOI 10.1016/0022-1236(84)90076-4
- Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. MR 997938
- Fu-Zhou Gong and Feng-Yu Wang, Functional inequalities for uniformly integrable semigroups and application to essential spectrums, Forum Math. 14 (2002), no. 2, 293–313. MR 1880915, DOI 10.1515/form.2002.013
- Fuzhou Gong and Liming Wu, Spectral gap of positive operators and applications, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 12, 983–988 (English, with English and French summaries). MR 1809440, DOI 10.1016/S0764-4442(00)01739-0
- Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
- Leonard Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups, Dirichlet forms (Varenna, 1992) Lecture Notes in Math., vol. 1563, Springer, Berlin, 1993, pp. 54–88. MR 1292277, DOI 10.1007/BFb0074091
- Masanori Hino, Exponential decay of positivity preserving semigroups on $L^p$, Osaka J. Math. 37 (2000), no. 3, 603–624. MR 1789439
- Shigeo Kusuoka, Analysis on Wiener spaces. II. Differential forms, J. Funct. Anal. 103 (1992), no. 2, 229–274. MR 1151548, DOI 10.1016/0022-1236(92)90121-X
- D. Revuz, Markov chains, North-Holland Mathematical Library, Vol. 11, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0415773
- Michael Röckner and Feng-Yu Wang, Weak Poincaré inequalities and $L^2$-convergence rates of Markov semigroups, J. Funct. Anal. 185 (2001), no. 2, 564–603. MR 1856277, DOI 10.1006/jfan.2001.3776
- O. S. Rothaus, Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities, J. Functional Analysis 42 (1981), no. 1, 102–109. MR 620581, DOI 10.1016/0022-1236(81)90049-5
- Barry Simon, A remark on Nelson’s best hypercontractive estimates, Proc. Amer. Math. Soc. 55 (1976), no. 2, 376–378. MR 400995, DOI 10.1090/S0002-9939-1976-0400995-2
- Barry Simon and Raphael Høegh-Krohn, Hypercontractive semigroups and two dimensional self-coupled Bose fields, J. Functional Analysis 9 (1972), 121–180. MR 0293451, DOI 10.1016/0022-1236(72)90008-0
- Feng-Yu Wang, Functional inequalities, semigroup properties and spectrum estimates, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 2, 263–295. MR 1812701, DOI 10.1142/S0219025700000194
- Feng-Yu Wang, Functional inequalities for the decay of sub-Markov semigroups, Potential Anal. 18 (2003), no. 1, 1–23. MR 1953493, DOI 10.1023/A:1020535718522
- Liming Wu, Uniformly integrable operators and large deviations for Markov processes, J. Funct. Anal. 172 (2000), no. 2, 301–376. MR 1753178, DOI 10.1006/jfan.1999.3544
- L. Wu, Essential spectral radius for Markov semigroups (I): discrete time case, Probab. Theory Relat. Fields 128 (2004), 255–321.
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
Additional Information
- Feng-Yu Wang
- Affiliation: Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China
- Email: wangfy@bnu.edu.cn
- Received by editor(s): October 15, 2002
- Received by editor(s) in revised form: June 3, 2003
- Published electronically: April 8, 2004
- Additional Notes: Supported in part by NNSFC(10025105, 10121101), TRAPOYT and the 973-Project
- Communicated by: Joseph A. Ball
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2629-2638
- MSC (2000): Primary 47D07, 60H10
- DOI: https://doi.org/10.1090/S0002-9939-04-07414-3
- MathSciNet review: 2054788