The commutant and similarity invariant of analytic Toeplitz operators on Bergman space

Abstract

The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-Wold Theorem does not hold in the Bergman space. In this paper, using the basis constructed by Michael and Zhu on the Bergman space we prove that each analytic Toeplitz operator M B(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. From the above theorem, we characterize the similarity invariant of some analytic Toeplitz operators by using K0-group term.

Similarity of analytic Toeplitz operators on the Bergman spaces

“…In this section we will show that two Toeplitz operators with symbols as finite Blaschke products are similar on the weighted Bergman spaces if and only if their symbols have the same order. This was conjectured in [6] and proved in [12] on the unweighted Bergman space. For another proof, see [7].…”

“…So the Bergman space is rigid. But in [12], Jiang and Li showed that if f is a finite Blaschke product, then the analytic Toeplitz operator T f is similar to the direct sum of finite copies of the Bergman shift T z on the unweighted Bergman space A 2 0 . In this paper, we will completely determine when two Toeplitz operators with symbol analytic on the closure of the unit disk are similar on the weighted Bergman spaces in terms of symbols, which is analogous to the result on the Hardy space [3].…”

“…, M n }. In[12], Jiang and Li showed that the Bergman space A 2 0 is the Banach direct sum M 1 M 2 · · · M n . The following theorem extends Jiang and He's result to the weighted Bergman spaces, which immediately gives Theorem 2.1.…”

Multiplication Operators on the Bergman Space

“…For the commutants of multiplication operators, we call the reader's attention to [T1,T2,T3,T4,Cow1,Cow2,Cow3], and also [AC,ACR,AD,CDG,CGW,Cl,Cu,GW1,JL,Ro,SZ1,SZ2,Zhu1,Zhu2]. On the Hardy space, [T1, T2] gave the characterization of commutant of multiplication operator with finite Blaschke product symbol.…”

On Similarity of Multiplication Operator on Weighted Bergman Space

“…In [10], J. Y. Hu, S. H. Sun, X. M. Xu and D. H. Yu proved that the analytic Toeplitz operator with finite Blaschke product symbol on the Bergman space has at least a reducing subspace on which the restriction of the associated Toeplitz operator is unitary equivalent to Bergman shift. In 2007, Jiang and Li [11] proved that each analytic Toeplitz operator M B(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. In this paper, we prove that multiplication operator M z n is similar to n 1 M z on the weighted Bergman space…”