| Title:
|
The spectra of general differential operators in the direct sum spaces (English) |
| Author:
|
Ibrahim, Sobhy El-sayed |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
9-29 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, the general ordinary quasi-differential expression $M_p$ of $n$-th order with complex coefficients and its formal adjoint $M_p^+$ on any finite number of intervals $I_p=(a_p,b_p)$, $p=1,\dots ,N$, are considered in the setting of the direct sums of $L_{w_p}^2(a_p,b_p)$-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new. (English) |
| Keyword:
|
quasi-differential expressions |
| Keyword:
|
essential spectra |
| Keyword:
|
joint field of regularity |
| Keyword:
|
regularly solvable operators |
| Keyword:
|
direct sum spaces |
| MSC:
|
34A05 |
| MSC:
|
34B24 |
| MSC:
|
34B25 |
| MSC:
|
34L05 |
| MSC:
|
34L15 |
| MSC:
|
47A10 |
| MSC:
|
47A55 |
| MSC:
|
47E05 |
| idZBL:
|
Zbl 1058.34110 |
| idMR:
|
MR2040216 |
| . |
| Date available:
|
2009-09-24T11:09:15Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127862 |
| . |
| Reference:
|
[1] N. I. Akhiezer and I. M. Glazman: Theory of Linear Operators in Hilbert Space, Part I, Part II.Frederich Ungar Publishing Co., New York, 1961, 1963. |
| Reference:
|
[2] J. Chaudhurim and W. N. Everitt: On the spectrum of ordinary differential operators.Proc. Roy. Soc. Edinburgh 68A (1969), 95–119. |
| Reference:
|
[3] D. E. Edmunds and W. D. Evans: Spectral Theory and Differential Operators.Oxford University Press, 1987. MR 0929030 |
| Reference:
|
[4] W. D. Evans: Regularly solvable extension of non-self-adjoint ordinary differential operators.Proc. Roy. Soc. Edinburgh 97A (1984), 79–95. MR 0751179 |
| Reference:
|
[5] W. D. Evans and Sobhy E. Ibrahim: Boundary conditions for general ordinary differential operators and their adjoints.Proc. Roy. Soc. Edinburgh 114A (1990), 99–117. MR 1051610 |
| Reference:
|
[6] W. N. Everitt: Integrable square solutions of ordinary differential equations.Quart. J. Math. (Oxford) 14 (1963), 170–180. Zbl 0123.05001, MR 0151660, 10.1093/qmath/14.1.170 |
| Reference:
|
[7] W. N. Everitt and A. Zettl: Generalized symmetric ordinary differential expressions I, the general theory.Nieuw Arch. Wisk. 27 (1979), 363–397. MR 0553264 |
| Reference:
|
[8] W. N. Everitt: Sturm Liouville differential operators in direct sum spaces.Rocky Mountain J. Math. 16 (1986), 497–516. Zbl 0624.34020, MR 0862277, 10.1216/RMJ-1986-16-3-497 |
| Reference:
|
[9] W. N. Everitt and D. Race: Some remarks on linear ordinary quasi-differential equations.Proc. London Math. Soc. 54 (1987), 300–320. MR 0872809 |
| Reference:
|
[10] W. N. Everitt: Differential operators generated by a countable number of quasi-differential expressions on the real line.Proc. London Math. Soc. 61 (1992), 526–544. |
| Reference:
|
[11] Sobhy E. Ibrahim: The spectra of well-posed operators.Proc. Roy. Soc. Edinburgh 125A (1995), 1331–1348. MR 1363006 |
| Reference:
|
[12] Sobhy E. Ibrahim: The point spectra and regularity fields of non-self-adjoint quasi-differential operators.Rocky Mountain J. Math. 25 (1995), 685–699. MR 1336556, 10.1216/rmjm/1181072243 |
| Reference:
|
[13] Sobhy E. Ibrahim: On the essential spectra of regularly solvable operators in the direct sum spaces.Rocky Mountain J. Math. 29 (1999), 609–644. MR 1705477, 10.1216/rmjm/1181071653 |
| Reference:
|
[14] M. A. Naimark: Linear Differential Operators, Part I, Part II.English Edition, Frederich Ungar Publishing Co., New York, 1967, 1968. |
| Reference:
|
[15] D. Race: On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators.Proc. Royal Soc. of Edinburgh 85A (1980), 1–14. Zbl 0422.34027, MR 0566064 |
| Reference:
|
[16] D. Race: On the essential spectra of linear $2n$-th order differential operators with complex coefficients.Proc. Roy. Soc. Edinburgh 92 (1982), 65–75. MR 0667125 |
| Reference:
|
[17] D. Race: The theory of $J$-self-adjoint extensions of $J$-symmetric operators.J. Differential Equations 57 (1985), 258–274. MR 0788280, 10.1016/0022-0396(85)90080-4 |
| Reference:
|
[18] M. I. Visik: On general boundary problems for elliptic differential operators.Amer. Math. Soc. 24 (1963), 107–172. |
| Reference:
|
[19] A. Zettl: Formally self-adjoint quasi-differential operators.Rocky Mountain J. Math. 5 (1975), 453–474. MR 0379976, 10.1216/RMJ-1975-5-3-453 |
| . |
