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Bounded Block Operator Matrices: The Quadratic Numerical Range; Special Classes of Block Operator Matrices; Spectral Inclusion; Estimates of the Resolvent; Corners of the Quadratic Numerical Range; Schur Complements and Their Factorization; Block Diagonalization; Spectral Supporting Subspaces; Variational Principles for Eigenvalues in Gaps; J-Self-Adjoint Block Operator Matrices; The Block Numerical Range; Numerical Ranges of Operator Polynomials; Gershgorin's Theorem for Block Operator Matrices; Unbounded Block Operator Matrices: Relative Boundedness and Relative Compactness; Closedness and Closability of Block Operator Matrices; Spectrum and Resolvent; The Essential Spectrum; Spectral Inclusion; Symmetric and J-Symmetric Block Operator Matrices; Dichotomous Block Operator Matrices and Riccati Equations; Block Diagonalization and Half Range Completeness; Uniqueness Results for Solutions of Riccati Equations; Variational Principles; Eigenvalue Estimates; Applications in Mathematical Physics: Upper Dominant Block Operator Matrices in Magnetohydrodynamics; Diagonally Dominant Block Operator Matrices in Fluid Mechanics; Off-Diagonally Dominant Block Operator Matrices in Quantum Mechanics.