The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces

ABSTRACT

In this paper, we consider the general quasi-differential
expressions each of order *n* with complex coefficients and their formal
adjoints on the interval (*a*,*b*). It is shown in direct sum spaces of functions
defined on each of the separate intervals with the cases of one and two
singular end-points and when all solutions of the equation and its adjoint are in (the limit
circle case) that all well-posed extensions of the minimal operator have resolvents
which are HilbertSchmidt integral operators and consequently have a wholly
discrete spectrum. This implies that all the regularly solvable operators have
all the standard essential spectra to be empty. These results extend those of
formally symmetric expression studied in
[1-10] and those of general quasi-differential expressions in [11-19].

Cite this paper

S. Ibrahim, "The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces,"*Advances in Pure Mathematics*, Vol. 3 No. 4, 2013, pp. 415-429. doi: 10.4236/apm.2013.34060.

S. Ibrahim, "The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces,"

References

[1] N. I. Akhiezer and I. M. Glazman, “Theory of Linear Operators in Hilbert Space,” Frederich Unger Publishing Co., New York, 1963.

[2] M. N. Naimark, “Linear Differential Operators,” New York, Ungar, Part I, 1967, Part II, 1968.

[3] W. N. Everitt and A. Zettl, “The Number of Integrable Square Solutions of Products of Differential Expressions,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 76 A, 1977, pp. 215-226.

[4] W. N. Everitt and A. Zettl, “Generalized Symmetric Ordinary Differential Expressions I,” The General Theory. Niew Archief Voor Wiskunde, Vol. 1, No. 3, 1979, pp. 363-397.

[5] A. N. Krall and A. Zettl, “Singular Self-Adjoint Sturm-Liouville Problems,” Journal of Differential and Integral Equations, Vol. 1, No. 4, 1998, pp. 423-432.

[6] D. Race, “On the Location of the Essential Spectra and Regularity Fields of Complex Sturm-Liouville Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 85A, 1980, pp. 1-14. doi:10.1017/S0308210500011689

[7] D. Race, “On the Essential Spectra of Linear 2nd Order Differential Operators with Complex Coefficients,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 92A, 1982, pp. 65-75. doi:10.1017/S0308210500019934

[8] A. Zettl, “Deficiency Indices of Polynomials in Symmetric Differential Expressions,” II, Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 73A, No. 20, 1974 (1975), pp. 301-306.

[9] A. Zettl, “Formally Self-Adjoint Quasi-Differential Operators,” Rocky Mountain Journal of Mathematics, Vol. 5, No. 3, 1975, pp. 453-474. doi:10.1216/RMJ-1975-5-3-453

[10] S. E Ibrahim, “On the Products of Self-Adjoint SturmLiouville Differential Operators in Direct Sum Spaces,” Journal of Informatics and Mathematical Sciences, Vol. 4 No. 1, 2012, pp. 93-109.

[11] D. E. Edmunds and W. D. Evans, “Spectral Theory and Differential Operators,” Oxford University Press, Oxford, 1987.

[12] W. D. Evans, “Regularly Solvable Extensions of Non-Self-Adjoint Ordinary Differential Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 114 A, 1990, pp. 99-117.

[13] W. D. Evans and S. E. Ibrahim, “Boundary Conditions for General Ordinary Differential Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 97A 1984, pp. 79-95. doi:10.1017/S0308210500031851

[14] W. N. Everitt and D. Race, “Some Remarks on Linear Ordinary Quasi-Differential Expressions,” Journal of London Mathematical Society, Vol. 3, No. 54, 1987, pp. 300-320.

[15] S. E. Ibrahim, “Non-Self-Adjoint Quasi-Differential Operators with Discrete Spectra,” Rocky Mountain Journal of Mathematics, Vol. 25, No. 3, 1995, pp. 1053-1348. doi:10.1216/rmjm/1181072204

[16] S. E. Ibrahim, “The Spectra of Well-Posed Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 125 A, 1995, pp. 1331-1348.

[17] S. E. Ibrahim, “The Point Spectra and Regularity Fields of Products of Quasi-Differential Operators,” Indian Journal of Pure and Applied Mathematics, Vol. 31, No. 6, 2000, pp. 747-665.

[18] S. E. Ibrahim, “On the Essential Spectra of General Differential Operators,” Italian Journal of Pure and Applied Mathematics Y, No. 9, 2001, pp. 45-67.

[19] S. E Ibrahim, “On the Essential Spectra for Products of the General Quasi-Differential Operators and Their Adjoints,” International Journal of Pure and Applied Mathematics, Vol. 70, No. 5, 2011, pp. 659-689.

[20] M. I. Visik, “On General Boundary Problems for Elliptic Differential Equations,” American Mathematical Society Transl, Vol. 2, No. 24, 1963, pp. 107-172.

[21] N. A. Zhikhar, “The Theory of Extension of J-Symmetric Operators,” Ukranin, Mat. Z. XI, Vol. 4, 1959, pp. 352-365.

[1] N. I. Akhiezer and I. M. Glazman, “Theory of Linear Operators in Hilbert Space,” Frederich Unger Publishing Co., New York, 1963.

[2] M. N. Naimark, “Linear Differential Operators,” New York, Ungar, Part I, 1967, Part II, 1968.

[3] W. N. Everitt and A. Zettl, “The Number of Integrable Square Solutions of Products of Differential Expressions,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 76 A, 1977, pp. 215-226.

[4] W. N. Everitt and A. Zettl, “Generalized Symmetric Ordinary Differential Expressions I,” The General Theory. Niew Archief Voor Wiskunde, Vol. 1, No. 3, 1979, pp. 363-397.

[5] A. N. Krall and A. Zettl, “Singular Self-Adjoint Sturm-Liouville Problems,” Journal of Differential and Integral Equations, Vol. 1, No. 4, 1998, pp. 423-432.

[6] D. Race, “On the Location of the Essential Spectra and Regularity Fields of Complex Sturm-Liouville Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 85A, 1980, pp. 1-14. doi:10.1017/S0308210500011689

[7] D. Race, “On the Essential Spectra of Linear 2nd Order Differential Operators with Complex Coefficients,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 92A, 1982, pp. 65-75. doi:10.1017/S0308210500019934

[8] A. Zettl, “Deficiency Indices of Polynomials in Symmetric Differential Expressions,” II, Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 73A, No. 20, 1974 (1975), pp. 301-306.

[9] A. Zettl, “Formally Self-Adjoint Quasi-Differential Operators,” Rocky Mountain Journal of Mathematics, Vol. 5, No. 3, 1975, pp. 453-474. doi:10.1216/RMJ-1975-5-3-453

[10] S. E Ibrahim, “On the Products of Self-Adjoint SturmLiouville Differential Operators in Direct Sum Spaces,” Journal of Informatics and Mathematical Sciences, Vol. 4 No. 1, 2012, pp. 93-109.

[11] D. E. Edmunds and W. D. Evans, “Spectral Theory and Differential Operators,” Oxford University Press, Oxford, 1987.

[12] W. D. Evans, “Regularly Solvable Extensions of Non-Self-Adjoint Ordinary Differential Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 114 A, 1990, pp. 99-117.

[13] W. D. Evans and S. E. Ibrahim, “Boundary Conditions for General Ordinary Differential Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 97A 1984, pp. 79-95. doi:10.1017/S0308210500031851

[14] W. N. Everitt and D. Race, “Some Remarks on Linear Ordinary Quasi-Differential Expressions,” Journal of London Mathematical Society, Vol. 3, No. 54, 1987, pp. 300-320.

[15] S. E. Ibrahim, “Non-Self-Adjoint Quasi-Differential Operators with Discrete Spectra,” Rocky Mountain Journal of Mathematics, Vol. 25, No. 3, 1995, pp. 1053-1348. doi:10.1216/rmjm/1181072204

[16] S. E. Ibrahim, “The Spectra of Well-Posed Operators,” Proceedings of the Royal Society of Edinburgh, Edinburgh, Vol. 125 A, 1995, pp. 1331-1348.

[17] S. E. Ibrahim, “The Point Spectra and Regularity Fields of Products of Quasi-Differential Operators,” Indian Journal of Pure and Applied Mathematics, Vol. 31, No. 6, 2000, pp. 747-665.

[18] S. E. Ibrahim, “On the Essential Spectra of General Differential Operators,” Italian Journal of Pure and Applied Mathematics Y, No. 9, 2001, pp. 45-67.

[19] S. E Ibrahim, “On the Essential Spectra for Products of the General Quasi-Differential Operators and Their Adjoints,” International Journal of Pure and Applied Mathematics, Vol. 70, No. 5, 2011, pp. 659-689.

[20] M. I. Visik, “On General Boundary Problems for Elliptic Differential Equations,” American Mathematical Society Transl, Vol. 2, No. 24, 1963, pp. 107-172.

[21] N. A. Zhikhar, “The Theory of Extension of J-Symmetric Operators,” Ukranin, Mat. Z. XI, Vol. 4, 1959, pp. 352-365.