# POLYNOMIALLY DEMICOMPACT OPERATORS AND SPECTRAL THEORY FOR OPERATOR MATRICES INVOLVING DEMICOMPACTNESS CLASSES

• Brahim, Fatma Ben (Department of Mathematics Faculty of Sciences of Sfax University of Sfax) ;
• Jeribi, Aref (Department of Mathematics Faculty of Sciences of Sfax University of Sfax) ;
• Krichen, Bilel (Department of Mathematics Faculty of Sciences of Sfax University of Sfax)
• Accepted : 2018.07.20
• Published : 2018.09.30

#### 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.

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