Multivariable Beurling–Lax representations: the commutative and free noncommutative settings

Abstract

The original theorem of Beurling asserts that any invariant subspace for the shift operator (multiplication by the coordinate function \( \chi(\lambda) =\lambda\)) on the Hardy space over the unit disk can be represented as an inner function times H2. We survey various approaches (including ideas and techniques from engineering systems theory and reproducing kernel Hilbert spaces) beyond Beurling’s original approach developed over the years for proving this result and then focus on understanding how these approaches can be adapted to handle the more delicate situation where the Hardy-space shift is replaced by the shift operator on a weighted Bergman space over the unit disk. We then indicate how all these results can be extended further to the setting of freely noncommutative shift-operator tuples on a weighted Bergman space in several freely noncommuting indeterminates.