الفهرس 
Definitions and notionsOnly 14 pages are availabe for public view
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Abstract
It is well known that the summation of the diagonal elements in a square matrix is equal to the summation of the eigenvalues of linear operator in finite dimensional space. In other words, the trace of a matrix is equal to the spectral trace in n-dimensional space. It is worth mentioning that this theorem is satisfied also in the case of unclear operators which are defined in Hillbert space [24]. Thus we might ask the following question : Is this theorem applicable in case of unbounded operators ? ; especially in the case of differential operators the trace of matrices and spectral trace are not exist. For this reason we define the socalled ”Regularized trace”. The study of regular trace for differential operators plays an important role in several fields such as mathematical analysis, theoretical physics and quantum mechanics. We can use also the regular trace in the inverse spectral problems in functional analysis. A good number of works has been devoted to the deduction of the formulae of regularized traces of differential operators such as L. M. Gelfand, B. M. Levitan [I], J. A. charls, JR. Halberg and V. A. Kramer [2], V. B. Lidsky, V. A. Sadovnichii [3], [4], V. A-Sadovnichii, V. A. Lyubishkin [5], Y. Belabbaci [6], S. A. Saleh [7], H. A. Zedan [8], S. A. Saleh, M. A. Kassem [9], S. A. Saleh, R. M. Allam [lo], V. A. Lyubishkin 1 1 A. S. Pechentsov [l2]-[l3], V. A. Sadovnichii, V. V. Dubrovskii [l4], [16], D. Milinkovic [15] and others. The concept of regularized sum for eigenfunction of differential operators are introduced by S. A. Saleh [18], S. A. Saleh R. M. Allam [19]. Also the proof of expansion theorem of eigenfunction for multi-point and integral conditions is given by S. A. Saleh, M. A. Kassem [20].