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The Well-Posed Operators with Their Spectra in L^pw-Spaces

Advances in Pure Mathematics 2023年第6期13卷 347-368页

作者： Sobhy El-Sayed Ibrahim Faculty of Basic Education Department of Mathematics Public Authority of Applied Education and Training Kuwait Kuwait

In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τp,q in Lpw-spaces of order n with complex coefficients and its formal adjoint τ+q',p' in Lpw-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T0 (τp,q) generated by such expression τp,q and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt 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. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.

关键词： quasi-differential expressions Regular and Singular Endpoints Minimal and Maximal Operators Regularly Solvable Operators Well-Posed Operators Deficiency Indices

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On the Domains of General Ordinary differential Operators in the Direct Sum Spaces

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Advances in Pure Mathematics 2022年第3期12卷 206-228页

作者： Sobhy El-Sayed Ibrahim Faculty of Basic Education Department of Mathematics Public Authority of Applied Education and Training Kuwait City Kuwait

Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [ap,bp). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.

关键词： quasi-differential expressions Regular and Singular Equations Minimal and Maximal Operators Regularly Solvable Operators J-Self-Adjoint Extension Boundary Conditions

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