Abstract
Perturbations of a Dirichlet form \(\mathfrak{h}\) by measures μ are studied. The perturbed form \(\mathfrak{h}\)−μ−+μ+ is defined for μ− in a suitable Kato class and μ+ absolutely continuous with respect to capacity. L p-properties of the corresponding semigroups are derived by approximating μ− by functions. For treating μ+, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has L p -L q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L 1 the same is shown to be true for the perturbed semigroup, for a large class of measures.
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S. Albeverio, J. Brasche and M. Röckner: ‘Dirichlet forms and generalized Schrödinger operators’, in Schrödinger Operators, Proceedings, Sonderborg, H. Holden and A. Jensen (eds.). Springer, Lecture Notes in Physics 345 (1989), 1–42.
S.Albeverio and Z.Ma: ‘Perturbation of Dirichlet forms-Lower semiboundedness, closability, and form cores’, J. Funct. Anal. 99 (1991), 332–356.
S. Albeverio and Z. Ma: ‘Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms’, Preprint SFB 237, 1989.
W.Arendt and C. J. K.Batty: Absorption semigroups and Dirichlet boundary conditions. Math Ann. 295 (1993), 427–448.
H.Bauer: Maβ- und Integrationstheorie, W. de Gruyter, Berlin, 1990.
J.Baxter, G.DalMaso and U.Mosco: Stopping times and Г-convergence. Trans. Amer. Math. Soc. 303 (1987), 1–38.
P.Blanchard and Z.Ma: Semigroup of Schrödinger operators with potentials given by Radon measures. In. Stochastic Processes-Physics and Geometry, S.Albeverio et al. (eds.), Proceedings, Ascona. World Sci., Singapore, 1990.
N.Bouleau and F.Hirsch: Dirichlet Forms and Analysis on Wiener Space. W. de Gruyter, Berlin, 1991.
J. Brasche, P. Exner, Yu. Kuperin and P. Seba: Schrödinger operators with singular interactions. SFB 237-Preprint Nr. 132, Bochum, 1991.
R.Carmona, W. Ch.Masters and B.Simon: Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91 (1990), 117–142.
T.Coulhon: ‘Dimension à l'infini d'un semi-groupe analytique’, Bull. Sc. Math. 2o série, 114 (1990), 485–500.
E. B.Davies. One-Parameter Semigroups. Academic Press, London, 1980.
E. B.Davies: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge, 1989.
M. Demuth: ‘On topics in spectral and stochastic analysis for Schrödinger operators’, Proceedings Recent Developments in Quantum Mechanics, A. Boutet de Mouvel et al. (eds.), Kluwer, 1991.
A.Devinatz: Schrödinger operators with singular potentials. J. Operator Theory 4 (1980), 25–35.
T.Figiel, T.Iwaniecz and A.Pelczyński: Computing norms of some operators in L p-spaces. Studia Math. 79 (1984), 227–274.
M.Fukushima. Dirichlet Forms and Markov Processes. North Holland, Amsterdam, 1980.
M.Fukushima, K.Sato, and S.Taniguchi: On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures. Osaka J. Math. 28 (1991), 517–535.
L. I. Hedberg: Spectral synthesis and stability in Sobolev spaces, in Euclidean Harmnic Analysis, Proceedings Maryland, J. J. Benedetto (ed.), Lect. Notes in Math. 779 (1980), 73–103.
I. W.Herbst and A. D.Sloan: Perturbation of translation invariant positivity preserving semigroups on L 2(R N). Trans. Amer. Math. Soc. 236 (1978), 325–360.
W.Hoh and N.Jacob: Some Dirichlet forms generated by pseudo differential operators. Bull. Sci. Math. 116 (1992), 383–398.
N.Jacob: Dirichlet forms and pseudo differential operators. Expo. Math. 6 (1988), 313–351.
T.Kato: Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1980.
T. Kato: ‘L p-theory of Schrödinger operators with a singular potential’, in Aspects of Positivity in Functional Analysis, Nagel, Schlotterbeck, Wolff (eds.), North-Holland, 1986.
Z.Ma and M.Röckner. An Introduction to the Theory of (Non-Symmetric) Dirichlet-Forms. Springer-Verlag, Berlin, 1992.
R.Nagel (ed.). One Parameter Semigroups of Positive Operators. Lect. Notes in Math. 1184, Springer-Verlag, Berlin, 1986.
E. M. Ouhabaz: ‘Propriétés d'ordre et de contractivité des semigroupes avec applications aux opérateurs elliptiques’, Thèse de doctorat, Besançon, 1992.
A.Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin, 1983.
M.Reed and B.Simon. Methods of Modern Mathematical Physics.I: Functional Analysis. Revised ed., Academic Press, New York, 1980.
M.Reed and B.Simon: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York, 1978.
F.Riesz and B.Sz.-Nagy. Vorlesungen über Funktionalanalysis. VEB Deutscher Verlag d. Wiss., Berlin, 1968.
H. H.Schaerfer: Banach Lattices and Positive Operators. Springer-Verlag, Berlin, 1974.
B.Simon: A canonical demposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28 (1978), 377–385.
B.Simon: Schrödinger semigroups. Bull. (N.S.) Amer. Math. Soc. 7 (1982), 447–526.
E. M.Stein. Topics in Harmonic Analysis, Related to the Littlewood-Paley Theory. Princeton Univ. Press, Princeton, N.J., 1970.
P. Stollmann: Formtechniken bei Schrödingeroperatoren. Diplom thesis Munich, 1985 (unpublished).
P.Stollmann: ‘Admissible and regular potentials for Schrödinger forms’, J. Operator Theory 18 (1987), 139–151.
P.Stollmann: ‘Smooth perturbations of regular Dirichlet forms’, Proc. Amer. Math. Soc. 116 (1992), 747–752.
P.Stollmann: Closed ideals in Dirichlet spaces. Potential Analysis 2 (1993), 263–268.
K.-T.Sturm: Measures charging no polar sets and additive functionals of Brownian motion. Forum Math. 4 (1992), 257–297.
K.-T. Sturm: Schrödinger operators with arbitrary nonnegative potentials, in Operator Calculus and Spectral Theory, Proc. Lambrecht 1991, Demuth, Gramsch, Schulze eds., Birkhäuser, 1992, 291–306.
N.Varopoulos: Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63 (1985), 240–260.
J.Voigt: On the perturbation theory for strongly continuous semigroups. Math. Ann. 229 (1977), 163–171.
J.Voigt: Absorption semigroups, their generators, and Schrödinger semigroups. J. Funct. Anal. 67 (1986), 167–205.
J.Voigt: Absorption semigroups. J. Operator Theory 20 (1988), 117–131.
J.Voigt: One-parameter semigroups acting simultaneously on different L p-spaces. Bull. Soc. Royale. Sc. Liège 61 (1992), 465–470.
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Stollmann, P., Voigt, J. Perturbation of Dirichlet forms by measures. Potential Analysis 5, 109–138 (1996). https://doi.org/10.1007/BF00396775
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DOI: https://doi.org/10.1007/BF00396775