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This book contains an up-to-date account of those parts of the theory of bounded and closed linear operators in Banach and Hilbert spaces, relevant to spectral problems involving differential equations. The book also looks at essential spectra, non-compactness, eigenvalues, entropy and approximation numbers. The abstract theory is illustrated by results for embedding maps between Sobolev spaces and strong emphasis is placed on application to boundary-value problems for general second-order linear elliptic equations in an arbitrary domain in Rn. The book provides a survey of the work that has been done in this area in recent years.