Editing Operator theory (section)

===Connection with complex analysis===
Many operators that are studied are operators on Hilbert spaces of [[holomorphic function]]s, and the study
of the operator is intimately linked to questions in function theory.
For example, [[Beurling's theorem]] describes the [[invariant subspace]]s 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]].<ref>{{citation|first=Nikolai|last=Nikolski|authorlink = Nikolai Nikolski|title=A treatise on the shift operator|publisher=Springer-Verlag|year=1986| isbn=0-387-90176-0}}. A sophisticated treatment of the connections between Operator theory and Function theory in the [[Hardy space]].</ref>  The success in studying multiplication operators, and more generally [[Toeplitz operator]]s (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]].