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Reciprocal transformations of Hamiltonian operators of hydrodynamic type areinvestigated. The transformed operators are generally nonlocal, possessing anumber of remarkable algebraic and differential-geometric properties. We applyour results to linearly degenerate semi-Hamiltonian systems in Riemanninvariants. Since all such systems are linearizable by appropriate(generalized) reciprocal transformations, our formulae provide an infinity ofmutually compatible nonlocal Hamiltonian structures, explicitly parametrized byarbitrary functions of one variable.Comment: 26 page

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems