## On unbounded integral operators with quasisymmetric kernels.(Russian. English summary)Zbl 1463.45058

Summary: In 1935 von Neumann established that a limit spectrum of self-adjoint Carleman integral operator in $$L_2$$ contains 0. This result was generalized by the author on nonself-adjoint operators: the limit spectrum of the adjoint of Carleman integral operator contains 0. Say that a densely defined in $$L_2$$ linear operator $$A$$ satisfies the generalized von Neumann condition if 0 belongs to the limit spectrum of adjoint operator $$A^{\ast}$$. Denote by $$B_0$$ the class of all linear operators in $$L_2$$ satisfying a generalized von Neumann condition. The author proved that each bounded integral operator, defined on $$L_2$$, belongs to $$B_0$$. Thus, the question arises: is an analogous assertion true for all unbounded densely defined in $$L_2$$ integral operators? In this note, we give a negative answer on this question and we establish a sufficient condition guaranteeing that a densely defined in $$L_2$$ unbounded integral operator with quasisymmetric lie in $$B_0$$.

### MSC:

 45P05 Integral operators 47B34 Kernel operators
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### References:

 [1] Korotkov V. B., “On Some Properties of Partially Integral Operators”, Dokl. Akad. Nauk SSSR, 217:4 (1974), 752-754 (in Russian) · Zbl 0302.47038 [2] Korotkov V. B., Integral Operators, Izd-vo Novosib. Gos. Un-ta, Novosibirsk, 1977, 68 pp. (in Russian) [3] Halmos P. R., Sunder V. S., Bounded Integral Operators on $$L^2$$ Spaces, Springer Verlag, Berlin-Heidelberg-New York, 1978, 134 pp. · Zbl 0389.47001 [4] Korotkov V. B., “On One Class of Linear Operators in $$L_2$$”, Siberian Mathematical Journal, 60:1 (2019), 89-92 · Zbl 07079963 [5] Korotkov V. B., Integral Operators, Nauka, Novosibirsk, 1983, 224 pp. (in Russian) [6] Korotkov V. B., “On Partially Measure Compact Unbounded Linear Operators in $$L_2$$”, Vladikavkaz. Math. J., 18:1 (2016), 36-41 (in Russian) · Zbl 1474.47054 [7] Korotkov V. B., “Integral Equations of the Third Kind with Unbounded Operators”, Siberian Mathematical Journal, 58:2 (2017), 255-263 · Zbl 1372.45002
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