Fractional powers of the algebraic sum of normal operators

Diagana, Toka

doi:10.1090/S0002-9939-05-08183-9

Fractional powers of the algebraic sum of normal operators
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Proc. Amer. Math. Soc. 134 (2006), 1777-1782 Request permission

Abstract:

The main concern in this paper is to give sufficient conditions such that if $A, B$ are unbounded normal operators on a (complex) Hilbert space $\mathbb H$, then for each $\alpha \in (0 , 1)$, the domain $D((\overline {A+B})^{\alpha })$ equals $D(A^{\alpha }) \cap D(B^{\alpha })$. It is then verified that such a result can be applied to characterize the domains of fractional powers of a large class of the Hamiltonians with singular potentials arising in quantum mechanics through the study of the Schrödinger equation.

References

António de Bivar-Weinholtz and Michel L. Lapidus, Product formula for resolvents of normal operators and the modified Feynman integral, Proc. Amer. Math. Soc. 110 (1990), no. 2, 449–460. MR 1013964, DOI 10.1090/S0002-9939-1990-1013964-6
Haïm Brézis and Tosio Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 (1979), no. 2, 137–151. MR 539217
Tocka Diagana, Sommes d’opérateurs et conjecture de Kato-McIntosh, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 461–464 (French, with English and French summaries). MR 1756959, DOI 10.1016/S0764-4442(00)00200-7
Toka Diagana, Variational sum and Kato’s conjecture, J. Convex Anal. 9 (2002), no. 1, 291–294. MR 1917402
Tocka Diagana, Quelques remarques sur l’opérateur de Schrödinger avec un potential complexe singulier particulier, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), no. 2, 293–298 (French, with French summary). MR 2017083
Toka Diagana, Algebraic sum of unbounded normal operators and the square root problem of Kato, Rend. Sem. Mat. Univ. Padova 110 (2003), 269–275. MR 2033011
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Tosio Kato, Fractional powers of dissipative operators. II, J. Math. Soc. Japan 14 (1962), 242–248. MR 151868, DOI 10.2969/jmsj/01420242
J.-L. Lions, Espaces intermédiaires entre espaces hilbertiens et applications, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.) 2(50) (1958), 419–432 (French). MR 151829
J.-L. Lions, Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs, J. Math. Soc. Japan 14 (1962), 233–241 (French). MR 152878, DOI 10.2969/jmsj/01420233
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Atsushi Yagi, Coïncidence entre des espaces d’interpolation et des domaines de puissances fractionnaires d’opérateurs, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 6, 173–176 (French, with English summary). MR 759225