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Báo cáo toán học: "A spectral characterization of monotonicity properties of normal linear operators with an aplication to nonlinear telegraph equation "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một đặc tính quang phổ của các thuộc tính monotonicity của các nhà khai thác tuyến tính bình thường với một ứng dụng phương trình phi tuyến điện báo. | J. OPERATOR THEORY 1 1 1984 333--341 Copyright by INCREST 1984 A SPECTRAL CHARACTERIZATION OF MONOTONICITY PROPERTIES OF NORMAL LINEAR OPERATORS WITH AN APPLICATION TO NONLINEAR TELEGRAPH EQUATION G. HETZER 1. Let H be a real Hilbert space and T H dom T - H be a densely defined closed linear operator with ran T ker T L. The inverse T-1of T dom T nran T is a bounded linear operator on ran T according to the open mapping theorem hence a T sup c I c 6 R 0 VxG dom T 7x x ỈS - c-illTxji2 is a positive real number or oo. The interest ina T arises from the important paper of Brezis and Nirenberg 3 on semilinear problems at resonance where this quantity serves as a measure of monotonicity for the linear part. Indeed we have a T oo iff T is monotone and in case of a T oo it is the largest positive number for which T-1 c-1Idran D is monotone on ran T . If T is selfadjoint and a T oo one readily sees that 0i T is the largest spectral value of T less than 0. This describes the relation between 3 and other papers e.g. 1 6 7 on resonance problems which start from spectral properties of the linear part. Moreover such a characterization turns out to be quite useful for determining a T in applications since the spectrum of various differential operators can be found in the literature. Here we treat the case where T is normal. A nonselfadjoint normal linear operator is induced e.g. by the telegraph operator. Of course this demands to include the complex part of the spectrum of T into consideration. To this end we denote the complexificationof H by Hc the elements of which are written as X 4- iy x y e H . Thus the linear structure is given by u V xa 4- x i y y and Ẳx Ịiyu i Ẳyu 4- px for u xu iy xu yueH V x 4- yv e H and ệ Ả 4- ip 2 peR . The inner product is defined by u v x x 4- y y 4- i y x - x y . 334 G. HETZER Moreover we associate a complex linear operator Ac Hc dom y4c - Hc with any linear operator A H dom A - H by setting dom 4c x iy X y e e dom 4 and Ac x iy -- Ax iAy for X