On Similarity and Reducing Subspaces of the n-Shift plus Certain Weighted Volterra Operator

Copyright © 2017 Yucheng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let g(z) be an n-degree polynomial (n ≥ 2). Inspired by Sarason’s result, we introduce the operator T1 defined by the multiplication operatorMg plus the weighted Volterra operatorVg on the Bergman space. We show that the operator T1 is similar toMg on some Hilbert space S2 g(D). Then for g(z) = zn, by using matrix manipulations, the reducing subspaces of the corresponding operator T2 on the Bergman space are characterized.


Introduction
The invariant subspace and reducing subspace problems are interesting and important themes in operator theory.The conjecture is that every bounded linear operator  on a separable Hilbert space  has a nontrivial closed invariant subspace.A closed linear nontrivial subspace  of  is called an invariant subspace for  if  is different from {0} and  such that  ⊂ .If  and  ⊥ are both invariant subspaces for , then  is said to be a reducing subspace for .The invariant subspace and reducing subspace problems on the Hardy space and the Bergman space have been studied extensively in the literature.We mention here the papers [1][2][3][4][5][6][7][8][9][10][11][12][13] and the books [14][15][16][17] which include a lot of the information on the corresponding operator theory.
Let D be the unit disk in the complex plane C, and let  2  (D) denote the Bergman space of analytic functions which belongs to  2 (D), where  2 (D) is the space of square integrable functions on D. It is well known that  2  (D) is a Hilbert space.If  ∈  2  (D), then where  is the normalized area measure on D, and For ,  ∈  2  (D), let and then the inner product of  and  is defined by In this inner product,  2  (D) has an orthonormal basis {  } ∞ =0 , where   () = √  + 1  , ( = 0, 1, . ..) . ( Let  ∞ (D) denote the algebra of bounded analytic functions on D. For  ∈  ∞ (D),   is an analytic multiplication operator on the Bergman space defined by is a bounded linear operator on Over the years it has been shown that many familiar classes of operators do have invariant subspaces.The lattice of shift operator acting on the Hardy space is completely described by Beurling's Theorem [4].Sarason (see [11]) characterized all closed invariant subspaces of the Volterra operator for  ∈  2 (0, 1) , 0 <  ≤  < 1.

(8)
In [1], Aleman characterized boundedness and compactness of the integral operator between Hardy space   and   for ,  > 0. In [2], using Beurling's Theorem, Aleman and Korenblum studied the complex Volterra operator in the Hardy space  2 (D) defined by Then they characterized the lattice of closed invariant subspaces of .Sarason (see [11]) studied the lattice of closed invariant subspaces of   +  acting on  2 (0, 1).Montes-Rodriguez et al. (see [9]) and Cowen et al. (see [5]) used the idea of Sarason to study the invariant subspaces of certain classes of composition operators on the Hardy space.Following Sarason's work, Čučković and Paudyal (see [6]) characterized the lattice of closed invariant subspaces of the shift plus complex Volterra operator on the Hardy space.In their paper, the operator  is defined by Ball (see [3]) and Nordgren (see [10]) studied the problem of determining the reducing subspaces for an analytic Toeplitz operator on the Hardy space.In [12], Stessin and Zhu gave a complete description of the weighted unilateral shift operator of finite multiplicity on some Hilbert spaces type I and type II.In [13], Zhu described the properties of the commutant of analytic Toeplitz operators with inner function symbols on the Hardy space and the Bergman space and characterized the reducing subspaces of a class of multiplication operators.In 2011, Douglas and Kim in [7] studied the reducing subspaces for an analytic multiplication operator    on the Bergman space  2  (  ) of the annulus   .Based on the above works, for an -degree polynomial () ( ≥ 2), we introduce the operator  1 defined by the multiplication operator   plus the weighted Volterra operator   on the Bergman space.We show that the operator  1 is similar to   on some Hilbert space  2  (D).Then for () =   , by using matrix manipulations, the reducing subspaces of corresponding operator  2 on the Bergman space are characterized.

The Similarity of the Operator 𝑇 1
For an -degree polynomial () ( ≥ 2), the operator  1 is defined by To prove our result, we introduce the space  2  (D) defined by where (D) is the space of holomorphic functions on the unit disk, and  = / is the differentiation operator.From the definition of So ℎ() ∈  2  (D).It can be shown that, for any holomorphic function ℎ with ℎ(0) = 0 and ℎ() ∈  2  (D), then ‖ℎ‖ 2 2 ≤ ‖ℎ‖ 2  2 .So the norm of  2  (D) is defined by Corresponding inner product is given by In the following, we suppose that   () has no zero points on D. This condition guarantees that  2  (D) is closed under the given norm.
(iii) The operator  1 acting on  2  (D) is similar under   to the multiplication operator   on  2  (D).
(ii) First we want to show   is a bounded operator on  2  (D).For  ∈  2  (D), we have Clearly   is linear.To show   is one to one, suppose that Differentiating both sides, we obtain that  1 () =  2 () and hence   is one to one.From the definition of   we have that   ((1/  ())ℎ)() = ℎ(), for ℎ ∈ Now applying   on both sides of the above equality, we obtain that So    1 =     and    1  −1  =   .That is to say,   transforms the operator  1 into the multiplication operator   on  2  (D).

The Reducing Subspaces of the Operator 𝑇 2
In this section, for fixed  ≥ 2, we consider the case of () =   .That is, the operator  2 is defined by for  ∈ with respect to the orthonormal basis , where   ∈ C (,  ≥ 1) and   = √  + 1.Moreover, if  is a projection, then  ∈ A  (  ) if and only if  =  or 0.

Proof. Denote the orthonormal basis of 𝐿
So the operator   admits the following matrix representation with respect to the above basis: Suppose that  has the following matrix representation with respect to the orthonormal basis of From    =   , we have ) . Thus So we obtain Conversely, if  admits the matrix representation (30) with respect to the above basis, simple computation shows that    =   .So  ∈ A  (  ).Moreover, if  is a projection, we deduce that  ∈ A  (  ) if and only if  has the following form: where  11 = 1 or 0.
Remark 3. In fact, since   is irreducible on  2  (D), any projection in A  (  ) is  or 0.
The following lemma will be used in the proof of main theorem.
(iii)   is a reducing subspace of  2 .
Proof.(i) and (ii) are obvious.We only need to show (iii).Note that So we have  2   ⊂   and as desired.
From [14], we know that determining the reducing subspaces of  2 is equivalent to finding the projection in the commutant of  2 .Thus we have the following conclusion.
(iii) The operator  2 acting on  2  (D) is similar under    to the multiplication operator    on  2  (D).