Abstract
Let A0, A1 be positive selfadjoint operators in a Hilbert space, let |·| be the norm in some symmetrically normed ideal of operators, satisjying the majorization condition. In the paper one obtains estimate
and also some of its generalizations. Examples, with A0, A1 differential operators, are considered.
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M. Sh. Birman, L. S. Koplienko, and M. Z. Solomyak, “Estimates of the spectrum of a difference of fractional powers of selfadjoint operators,” Izv. Vyssh. Uchebn. Zaved. Matematika, No. 3 (154), 3–10 (1975).
M. Sh. Birman and M. Z. Solomyak, “The spectral asymptotics of nonsmooth elliptic operators. II,” Trudy Moskov. Mat. Obshch.,28, 3–34 (1973).
M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden (1976).
M. Sh. Birman and M. Z. Solomyak, “Operator integration, perturbations, and commutators,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,170, 34–66 (1989).
V. I. Matsaev and Yu. A. Palant, “On the powers of a bounded dissipative operator,” Ukr. Mat. Zh.,14, No. 3, 329–337 (1962).
P. E. Sobolevskii, “Comparison theorems for fractional powers of operators,” Dokl. Akad. Nauk SSSR,174, No. 2, 294–297 (1967).
S. N. Naboko, “Estimates in symmetrically normed ideals for the difference of powers of accretive operators,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., Ser.1, No. 2, 40–45 (1988).
S. N. Naboko, Operator R-Functions and Their Applications in Spectral Analysis, Doctoral dissertation, Leningrad (1987).
I. Ts. Gokhberg (I. C. Gohberg) and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence (1969).
M. Sh. Birman and M. Z. Solomyak, “Double Stieltjes operator integrals,” Problemy Mat. Fiz., No. 1, 33–67 (1966).
M. Sh. Birman and M. Z. Solomyak, “Double Stieltjes operator integrals. III,” Problemy Mat. Fiz., No. 6, 27–53 (1973).
S. G. Krein, Yu. I. Petunin (Ju. I. Petunin), and E. M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence (1982).
M. Sh. Birman, “Perturbations of the continuous spectrum of a singular elliptic operator under the variation of the boundary and the boundary conditions,” Vestnik Leningrad. Univ., No. 1, 22–55 (1962).
M. Sh. Birman and M. Z. Solomyak, “The asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations,” Sib. Mat. Zh.,20, No. 1, 3–22 (1979).
G. Grubb, Functional Calculus of Pseudo-Differential Boundary Problems, Birkhäuser, Boston (1986).
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976).
V. G. Maz'ya, Sobolev Spaces [in Russian], Izd. Leningrad. Gos. Univ., Leningrad (1985).
V. G. Maz'ya and T. O. Shaposhnikova, Multipliers in Spaces of Differentiable Functions [in Russian], Leningrad State Univ., Leningrad (1986).
G. V. Rozenblyum, “Distribution of the discrete spectrum of singular differential operators,” Dokl. Akad. Nauk SSSR,202, No. 5, 1012–1015 (1972).
M. Sh. Birman and M. Z. Solomyak, “Quantitative analysis in Sobolev's imbedding theorems and applications to spectral theory,” in: Proc. Tenth Summer Math. School, 1972, Izd. Inst. Mat. Akad. Nauk USSR, Kiev (1974), pp. 5–189.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 120–145, 1989.
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Birman, M.S., Solomyak, M.Z. Estimates for the difference of fractional powers of selfadjoint operators in the case of unbounded perturbations. J Math Sci 61, 2018–2035 (1992). https://doi.org/10.1007/BF01095665
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DOI: https://doi.org/10.1007/BF01095665