Deficiency numbers of symmetric operators generated

Yu. M. Dyukarev

Sbornik. Mathematics, Tome 197 (2006) no. 8, pp. 1177-1203

Résumé

Symmetric block Jacobi matrices $J$ and the corresponding symmetric operators $L$ are studied. Let $m$ be the size of the blocks in the matrix $J$. As is known, the deficiency numbers $m_+$ and $m_-$ of the operator $L$ satisfy the inequalities $0\leqslant m_+,m_-\leqslant m$ and achieve their maximum value $m$ simultaneously. Let $m_+$ and $m_-$ be arbitrary integers such that $0\leqslant m_+,m_-\leqslant m-1$. It is shown that there exists a symmetric Jacobi matrix $J$ such that $m_+$ and $m_-$ are the deficiency numbers of the corresponding symmetric operator $L$. Bibliography: 13 titles.

Bibliographie

[1] M. G. Krein, “Beskonechnye $J$ matritsy i matrichnaya problema momentov”, Dokl. AN SSSR, 69:3 (1949), 125–128 | MR | Zbl

[2] M. G. Krein, “Osnovnye polozheniya teorii predstavleniya ermitovykh operatorov s indeksom defekta $(m,m)$”, Ukr. matem. zhurn., 1:2 (1949), 3–66 | MR | Zbl

[3] Yu. M. Berezanskii, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR | Zbl

[4] V. I. Kogan, “Ob operatorakh, porozhdennykh $l_p$-matritsami v sluchae maksimalnykh indeksov defekta”, Teoriya funktsii, funkts. analiz i ikh prilozh., 11, 1970, 103–107 | MR | Zbl

[5] A. G. Kostyuchenko, K. A. Mirzoev, “Priznaki vpolne neopredelennosti yakobievykh matrits s matrichnymi elementami”, Funkts. analiz i ego prilozh., 35:4 (2001), 32–37 | MR | Zbl

[6] I. V. Kovalishina, V. P. Potapov, “Radiusy kruga Veilya v matrichnoi zadache Nevanlinny–Pika”, Teoriya operatorov v funktsionalnykh prostranstvakh i ee prilozheniya, Sb. nauch. tr., Naukova dumka, Kiev, 1981, 25–49 | MR | Zbl

[7] V. K. Dubovoi, “Indefinitnaya metrika v interpolyatsionnoi probleme Shura dlya analiticheskikh funktsii. III”, Teoriya funktsii, funkts. analiz i ikh prilozh., 37, 1982, 14–26 | MR | Zbl

[8] V. I. Kogan, F. S. Rofe-Beketov, “On the question of the deficiency indices of differential operators with complex coefficients”, Proc. Roy. Soc. Edinburgh Sect. A, 72 (1975), 281–298 | MR | Zbl

[9] V. I. Kogan, F. S. Rofe-Beketov, “On square-integrable solutions of symmetric systems of differential equations of arbitrary order”, Proc. Roy. Soc. Edinburgh Sect. A, 74 (1976), 5–40 | MR | Zbl

[10] F. S. Rofe-Beketov, A. M. Kholkin, Spectral analysis of differential operators. Interplay between spectral and oscillatory properties, World Sci. Monogr. Ser. Math., 7, World Scientific, Hackensack, NJ, 2005 | MR | Zbl

[11] G. A. Kalyabin, “O chisle reshenii iz $L_2(0,\infty)$ samosopryazhennoi sistemy differentsialnykh uravnenii vtorogo poryadka”, Funkts. analiz i ego prilozh., 6:3 (1972), 74–76 | MR | Zbl

[12] V. P. Potapov, “Drobno-lineinye preobrazovaniya matrits”, Issledovaniya po teorii operatorov i ikh prilozheniyam, Sb. nauch. tr., Naukova dumka, Kiev, 1979, 75–97 | MR | Zbl

[13] N. I. Akhiezer, Klassicheskaya problema momentov i nekotorye voprosy analiza, svyazannye s neyu, GIFML, M., 1961 | MR | Zbl