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Unified Min-Max and Interlacing Theorems for Linear Operators

Catarina Araújo de Santa-Clara Gomes

There exist striking analogies between eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely: diagonalization theorems, interlacing inequalities and Courant-Fisher-type formulas. D. Carlson and E. Marques de Sá (Generalized minimax and interlacing inequalities, Linear and Multilinear Algebra, 15 (1984), 77-103) introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite dimensional case. In this paper, it is shown that this unification can be done using modular lattices with Goldie dimension (the Goldie dimension in Module Theory is naturally generalized to Lattice Theory - the Goldie dimension of a module being the Goldie dimension of the lattice of its submodules). Modular lattices have a natural structure of s-space in t he finite dimensional case. We are able to extend the unification of the results mentioned above to the countable (infinite) dimensional case.