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Báo cáo toán học: "Fredholm theory of piecewise continuous Fourier integral operators on Hilbert space "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Fredholm lý thuyết của các nhà khai thác piecewise liên tục Fourier tách rời trên không gian Hilbert. | J. OPERATOR THEORY 7 1982 51-60 Copyright by INCREST 1982 FREDHOLM THEORY OF PIECEWISE CONTINUOUS FOURIER INTEGRAL OPERATORS ON HILBERT SPACE s. c. POWER Duducava has studied the Fredholm theory of certain convolution integral operators on LP R and Z R c n of a very general kind and which are determined by piecewise continuous functions 6 7 . For example if denotes the Fourier transform then Fredholm criteria and an index theorem are obtained for the operators I Í-1 where ph Oị are piecewise continuous where My denotes multiplication by p and where Dy denotes the Fourier multiplier F MyF. It is easily checked that special cases of A include convolution integral operators singular integral operators and when 0 - 1 for z 1 2 .n the Fourier integral operator 2 Op a -i- a x ý éxyFf y áy 2tt J n with symbol function a x y PiWiAi . Z 1 In the present paper we concentrate on the Hilbert space case and the operators op ứ . By restricting to certain locally simple symbols whose operators behave locally like Hermitian operators we see that the essential spectrum of Op a is a line segment augmentation of the asymptotic range of a x ỳ and thus a finite union of closed curves. A natural winding number provides the Fredholm index. This is an explicit generalization of the continuous case 2 and although implicitly present it is by no means apparent in 6 7 . Being on a Hilbert space we can employ C -techniques and obtain further information about the generated pseudo-differential c -algebra. For example the character space can be determined and the corresponding character spectrum for Op rz identified with the asymptotic range of a x y . This is also shown to be 52 s. c. POWER true when Pi and Ipi i 1 2 . n are piecewise slowly oscillating. Our main technique which replaces the local principle of Gohberg and Krupnik employed by Duducava is to use Douglas s localization theorem for a c -algebra with centre and to characterize the local algebras as K2 0 1 the 2x2 matrices over .