Restriction of Toeplitz Operators on Their Reducing Subspaces

We study the restrictions of analytic Toeplitz operator on its minimal reducing subspaces for the unit disc and construct their models on slit domains. Furthermore, it is shown that is similar to the sum of copies of the Bergman shift.


Introduction
Let  be a bounded linear operator on a Hilbert space H; a subspace M of H is called an invariant subspace of  if (M) ⊆ M and a reducing subspace of  if M is an invariant subspace of  and  * .A reducing subspace M of  is called minimal if for every reducing subspace M of  such that M ⊆ M then either M = M or M = 0.For a concrete bounded operator  on a separable Hilbert space H, it is important to determine invariant subspaces and reducing subspaces for .
Let D be the unit open disc in the complex plane and () be the normalized area measure on D. The Bergman space  2  (D) consists of all analytic functions in the Lebesgue space  2 (()).It is clear that  2  (D) is a closed subspace of  2 (()), and let  denote the projection from  2 (()) onto  2  (D).The Toeplitz operator   on  2  (D) with symbol  ∈  ∞ (()) is defined by (  )() = ()(); it is called an analytic Toeplitz operator if  ∈  ∞ (D).
An th-order Blaschke product  is the analytic function on D given by where  is a real number and   ∈ D for 1 ≤  ≤ .A Blaschke product is very important in the theory of Hardy space.Characterization of reducing subspaces of an analytic Toeplitz operator   on Bergman space has been of great interest for last two decades.Thomson [1,2] showed that it suffices to study reducing subspace of   for a finite Blaschke product in the case of Hardy space.It can be generalized to Bergman spaces easily.
Zhu studied the reducing subspaces of   for a Blaschke product of order 2 firstly and showed that   has exactly two distinct minimal reducing subspaces (cf.[3]).Motivated by this fact, Zhu conjectured that the number of minimal reducing subspaces of   equals the order of  (cf.[3]).Guo et al. showed that in general this is not true (cf.[4]), and they found that the number of minimal reducing subspaces of   equals the number of connected components of the Riemann surface of () = () when the order of  is 3, 4, 6.Then they conjectured that the number of minimal reducing subspaces of   equals the number of connected components of the Riemann surface of () = () for any finite Blaschke product (called the refined Zhu's conjecture, cf.[4]).Douglas et al. confirmed the conjecture in [5,6] by using local inverses of Blaschke products [7].Tikaradze [8] generalized a part of results in [5] to bounded smooth pseudoconvex domains in C  .Douglas and Kim [9] studied reducing subspace    on Bergman space of the annulus; the case of Hardy space was summarized in [10].
In [11], Douglas et al. generalized the bundle shift [12] to the case of Bergman spaces, constructed a vector bundle model for analytic Toeplitz operator   on the Bergman space  2  (D), and tried to build vector bundle models for restrictions of   to its minimal reducing subspaces, but it is not completed.Douglas [13] studied unitary equivalence of the restrictions by computing their curvatures of corresponding

Models for Restriction of Toeplitz Operators on Their Minimal Reducing Subspaces
3.1.The Bergman Spaces on the Slit Disc.The domain  = D \ [0, 1) is called the slit disk.Let  denote the normalized area measure on . 2  () is the set of analytic functions in the Lebesgue space  2 (, ).For a nonnegative measurable function () on , we can define the weighted Bergman space with respect to ()() to be the set of all analytic functions in the Lebesgue space  2 (, ()).Ross studied invariant subspaces of Bergman spaces on slit domains in [16].Aleman et al. defined and studied the Hardy space of a slit domain and in particular they studied the invariant subspace of the slit disk; one can consult [17] for details.

Weighted Shift Models
Proposition 8.The restrictions of   2 on its minimal reducing subspaces are one-side weighted shifts, and they are not unitarily equivalent to each other.

2. Unitary Equivalence and Similarity of Weighted Shifts
{  } of complex numbers such that   =    +1 .Similarly,  is called a two-side weighted shift if there exist an orthonormal basis {  },  = . . ., −2, −1, 0, 1, 2, . . .for H and a bounded sequence {  } of complex numbers such that   =    +1 for all  ∈ Z. Suppose that  and  are two injective one-side weighted shifts with weights {  } and {  } respectively; then  is unitarily equivalent to  if and only if |  | = |  | for all .
Models for   2 .It is easy to check that 1/4|| is a measurable function on  and that (1/4||)() is a probability measure on .