On spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics

Rastegaev, N.

doi:10.1090/spmj/1525

On spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics
HTML articles powered by AMS MathViewer

by N. V. Rastegaev
Translated by: THE AUTHOR

St. Petersburg Math. J. 29 (2018), 1007-1029

DOI: https://doi.org/10.1090/spmj/1525

Published electronically: September 4, 2018

PDF | Request permission

Abstract:

The spectral asymptotics of a tensor product of compact operators in Hilbert space with known marginal asymptotics is studied. The methods of A. Karol′, A. Nazarov, and Ya. Nikitin are generalized to operators with almost regular marginal asymptotics. In many (but not all) cases it is shown that the tensor product in question also has almost regular asymptotics. The results are then applied to the theory of small ball probabilities of Gaussian random fields.

References

Siegfried Graf, Harald Luschgy, and Gilles Pagès, Functional quantization and small ball probabilities for Gaussian processes, J. Theoret. Probab. 16 (2003), no. 4, 1047–1062 (2004). MR 2033197, DOI 10.1023/B:JOTP.0000012005.32667.9d
Harald Luschgy and Gilles Pagès, Sharp asymptotics of the functional quantization problem for Gaussian processes, Ann. Probab. 32 (2004), no. 2, 1574–1599. MR 2060310, DOI 10.1214/009117904000000324
A. Papageorgiou and G. W. Wasilkowski, On the average complexity of multivariate problems, J. Complexity 6 (1990), no. 1, 1–23. MR 1048027, DOI 10.1016/0885-064X(90)90009-3
Andrei Karol′, Alexander Nazarov, and Yakov Nikitin, Small ball probabilities for Gaussian random fields and tensor products of compact operators, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1443–1474. MR 2357702, DOI 10.1090/S0002-9947-07-04233-X
Andrei I. Karol’ and Alexander I. Nazarov, Small ball probabilities for smooth Gaussian fields and tensor products of compact operators, Math. Nachr. 287 (2014), no. 5-6, 595–609. MR 3193939, DOI 10.1002/mana.201100010
Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR 1243717
M. Solomyak and E. Verbitsky, On a spectral problem related to self-similar measures, Bull. London Math. Soc. 27 (1995), no. 3, 242–248. MR 1328700, DOI 10.1112/blms/27.3.242
A. I. Nazarov, Logarithmic asymptotics of small deviations for some Gaussian processes in the $L_2$-norm with respect to a self-similar measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 311 (2004), no. Veroyatn. i Stat. 7, 190–213, 301 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 133 (2006), no. 3, 1314–1327. MR 2092208, DOI 10.1007/s10958-006-0041-x
G. N. Sytaja, Certain asymptotic representations for a Gaussian measure in Hilbert space, Theory of random processes, No. 2 (Russian), Izdat. “Naukova Dumka”, Kiev, 1974, pp. 93–104, 140 (Russian). MR 0362471
M. A. Lifshits, Asymptotic behavior of small ball probabilities, Prob. Theory and Math. Stat., VSP/TEV, Vilnius, 1999, pp. 453–468.
W. V. Li and Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, Handbook of Statist., vol. 19, North-Holland, Amsterdam, 2001, pp. 533–597. MR 1861734, DOI 10.1016/S0169-7161(01)19019-X
Small deviations for stochastic processes and related topics, Internet site, www.proba.jussieu.fr/ pageperso/smalldev/.
E. Csáki, On small values of the square integral of a multiparameter Wiener process, Statistics and probability (Visegrád, 1982) Reidel, Dordrecht, 1984, pp. 19–26. MR 758997
Wenbo V. Li, Comparison results for the lower tail of Gaussian seminorms, J. Theoret. Probab. 5 (1992), no. 1, 1–31. MR 1144725, DOI 10.1007/BF01046776
Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
A. A. Vladimirov and I. A. Sheĭpak, On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight, Funktsional. Anal. i Prilozhen. 47 (2013), no. 4, 18–29 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 47 (2013), no. 4, 261–270. MR 3185121, DOI 10.1007/s10688-013-0033-9
A. A. Vladimirov, An oscillation method in a problem on the spectrum of a fourth-order differential operator with self-similar weight, Algebra i Analiz 27 (2015), no. 2, 83–95 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 27 (2016), no. 2, 237–244. MR 3444462, DOI 10.1090/spmj/1385
N. V. Rastegaev, On spectral asymptotics of the Neumann problem for the Sturm-Liouville equation with self-similar weight of generalized Cantor type, J. Math. Sci. (N.Y.) 210 (2015), no. 6, 814–821. MR 3407794, DOI 10.1007/s10958-015-2592-1
David Damanik, Anton Gorodetski, and Boris Solomyak, Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian, Duke Math. J. 164 (2015), no. 8, 1603–1640. MR 3352042, DOI 10.1215/00127094-3119739
John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
Alexander Nazarov, Log-level comparison principle for small ball probabilities, Statist. Probab. Lett. 79 (2009), no. 4, 481–486. MR 2494639, DOI 10.1016/j.spl.2008.09.021
V. M. Zolotarev, Gaussian measure asymptotic in $l_2$ on a set of centered spheres with radii tending to zero, Proc. 12th Europ. Meeting of Statisticians, Varna, 1979, pp.254.
V. M. Zolotarev, Asymptotics of a Gaussian measure in $l_2$, Stability problems for stochastic models (Russian) (Moscow, 1984) Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, 1984, pp. 54–58 (Russian). MR 859211
J. Hoffmann-Jørgensen, L. A. Shepp, and R. M. Dudley, On the lower tail of Gaussian seminorms, Ann. Probab. 7 (1979), no. 2, 319–342. MR 525057
I. A. Ibragimov, The probability of a Gaussian vector with values in a Hilbert space hitting a ball of small radius, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979), 75–93, 239, 245 (Russian, with English summary). Investigations in the theory of probability distributions, IV. MR 535459
M. A. Lifshits, On the lower tail probabilities of some random series, Ann. Probab. 25 (1997), no. 1, 424–442. MR 1428515, DOI 10.1214/aop/1024404294