Abstract
For a linear operatorS in a Hilbert space ℋ, the relationship between the following properties is investigated: (i)S is singular (= nowhere closable), (ii) the set kerS is dense in ℋ, and (iii)D(S)∩ℛ(S)={0}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Albeverio, W. Karwowski, and V. Koshmanenko, “Square power of singularly perturbed operators,”Math. Nachr., 173, 5–24(1995).
W. Karwowski and V. Koshmanenko, “Additive regularization of singular bilinear forms,”Ukr. Mat. Zh.,42, No. 9, 1199–1204 (1990).
W. Karwowski and V. Koshmanenko, “On the definition of singular bilinear forms and singular linear operators,”Ukr. Mat. Zh.,45, No. 8, 1084–1089 (1993).
T. Kato,Theory of Perturbation of Linear Operators, Springer-Verlag, New York 1966.
V. D. Koshmanenko, “Perturbations of self-adjoint operators by singular bilinear forms,”Ukr. Mat. Zh.,41, No. 1, 3–18 (1989).
V. D. Koshmanenko, “Operator representation for nonclosable quadratic forms and the scattering problem,”Dokl. Akad. Nauk SSSR,20, 294–297 (1979).
V. D. Koshmanenko, “Singular perturbations defined by forms. Applications of self-adjoint extensions in quantum physics,”Lect.Notes. Phys.,324, 55–66 (1987).
S. Ota, “On a singular part of an unbounded operator,”Zeitschrift fur Anal Anwend.,7, 15–18 (1987).
B. Simon, “A canonical decomposition for quadratic forms with applications to monotone convergence theorems,”J. Fund. Anal.,28, 377–385 (1978).
V. D. Koshmanenko,Singular Bilinear Forms in Perturbations Theory of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev 1993.
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden,Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin (1988).
S. Albeverio, J. F. Brasche, and M. Rockner, “Dirichlet forms and generalized Schrodinger operators,”Lect. Notes Phys.,345, 1–42 (1989).
S. Albeverio, J. E. Fenstad, R. Hoegh-Krohn, W. Karwowski, and T. Lindström,Schrodinger Operators with Potentials Supportedby Null Sets, Preprint No. 8, SFB 237 (1991).
F. A. Berezin and L. D. Faddeev, “A remark on Schrodinger equation with singular potential,”Dokl Akad. Nauk SSSR,137, No. 5, 1011–1014 (1961).
M. Gorzelanczyk,Scattering Theory for the Singularly Perturbed Hamiltonians, Preprint, SFB 237 (1994).
W. Karwowski and V. D. Koshmanenko, “Regular restrictions of singular quadratic forms,”Funct. Anal. Appi,29, No. 2, 36–137 (1995).
V. D. Koshmanenko, “To the rank-one singular perturbations of self-adjoint operators,”Ukr. Mat. Zh.,43, No. 11, 1559–1566 (1991).
V. D. Koshmanenko,Dense Subspaces in the A-Scale of Hilbert Spaces, Preprint No. 835, ITP UWr (1993).
K. Schmudgen, “On domain of power of closed symmetric operator,”J. Oper. Theory,7, 53–75 (1983).
Yu. M. Berezanskii,Expansion in Eigenfunctions of Self-Adjoint Operators, AMS, Providence 1968.
Yu. M. Berezanskii,Self-Adjoint Operators in Spaces of Functions of Infinitely Many Variables, AMS, Providence 1986.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Koshmanenko, V.D., ôta, S. On characteristic properties of singular operators. Ukr Math J 48, 1677–1687 (1996). https://doi.org/10.1007/BF02529489
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02529489