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 Polynomially demicompact operators and spectral theory for operator matrices involving demicompactness classes Bull. Korean Math. Soc. 2018 Vol. 55, No. 5, 1351-1370 https://doi.org/10.4134/BKMS.b170775Published online September 30, 2018 Fatma Ben Brahim, Aref Jeribi, Bilel Krichen University of Sfax, University of Sfax, University of Sfax Abstract : In the first part of this paper we show that, under some conditions, a polynomially demicompact operator can be demicompact. An example involving the Caputo fractional derivative of order $\alpha$ is provided. Furthermore, we give a refinement of the left and the right Weyl essential spectra of a closed linear operator involving the class of demicompact ones. In the second part of this work we provide some sufficient conditions on the inputs of a closable block operator matrix, with domain consisting of vectors which satisfy certain conditions, to ensure the demicompactness of its closure. Moreover, we apply the obtained results to determine the essential spectra of this operator. Keywords : matrix operator, demicompact linear operator, Fredholm and semi-Fredholm operators, essential spectra MSC numbers : 47A53, 47A55, 47A10 Full-Text :