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The monograph is devoted to the study of spectral properties of energy operators whose elements are linear operators acting in Hilbert space. Fourth-order elliptic operators, in particular the bigarmonic operators, play a central role in a wide class of physical models such as linear elasticity theory, stiffness problems, image compression with optimized and sufficiently sparse stored data, and in the formulation of Stokes flows. In this connection, in solid state physics, quantum field theory, elasticity theory and many other fields, special attention is paid to the study of essential and discrete spectra of the energy operator, i.e., the Schrödinger-type biharmonic operator.