linguistic:translation, structural:reformatting
Connection with complex analysis

Many operators that are studied are operators on Hilbert spaces of holomorphic functions, and the studyof the operator is intimately linked to questions in function theory.For example, Beurling's theorem describes the invariant subspaces of the unilateral shift in terms of inner functions, which are bounded holomorphic functions on the unit disk with unimodular boundary values almost everywhere on the circle. Beurling interpreted the unilateral shift as multiplication by the independent variable on the Hardy space. The success in studying multiplication operators, and more generally Toeplitz operators (which are multiplication, followed by projection onto the Hardy space) has inspired the study of similar questions on other spaces, such as the Bergman space.