Wandering subspaces and the Beurling type Theorem I

Abstract

An elementary proof of the Aleman, Richter and Sundberg theorem concerning the invariant subspaces of the Bergman space is given. (2000). Primary 47A15, 32A35; Secondary 47B35. Mathematics Subject Classification

Random weighted shifts

“…The Bergman case was reproved by Sun-Zheng in [95] by lifting the Bergman space to the bidisk Hardy space. This proof was simplified by K. Izuchi [64]. If α < −5, examples of Hedenmalm-Zhu [55] disproves the property.…”

Beurling‐Type Theorem for a Class of Reproducing Kernel Hilbert Spaces

“…Let D be the open unit disk in the complex plane C. e Hardy space H 2 consists of all analytic functions on D having power series representations with ℓ 2 -complex coefficients sequence. at is (5) where H(D) is the space of analytic functions in D. e norm of the vector…”

“…is reveals the internal structure of invariant subspaces of the Bergman space and becomes a fundamental theorem on the Bergman space (see [4]). Later, the Beurling-type theorem was studied by many mathematicians (see [5][6][7][8][9]). In [8], Shimorin studied the Beurling-type theorem for a nonisometric operator T which is close to an isometry in some sense (in particular, we can assume T is left invertible), and proved the following theorem.…”

$$N_{\psi }$$-Type Quotient Modules in $$H^{2}({\mathbb {D}}^{n})$$

“…Let ψ(w) be an inner function on D, let K be the submodule generated by {(z 1 − ψ(z 2 )) f : f ∈ H 2 (D 2 )}. Izuchi and Yang [11] studied the spectrum, the selfcommutator and cross-commutator of the quotient module N ψ = H 2 (T 2 ) K. The Beurling type theorem was proved by Izuchi et al [9] for the quotient module N ψ . Wu and Xu [15] studied the reducing subspaces of these quotient modules.…”