Abstract
Let be the algebra generated in Sobolev space by the rational functions with poles outside the unit disk . In this paper, we study the similarity invariant of the multiplication operators in , when is univalent analytic on or is strongly irreducible. And the commutants of multiplication operators whose symbols are composite functions, univalent analytic functions, or entire functions are studied.
1. Introduction
Let be an analytic Cauchy domain in the complex plane and let denote the Sobolev space, denotes the planar Lebesgue measure. For , we define Then is a Hilbert space and a Banach algebra with identity under an equivalent norm. can be continuously embedded in the space of continuous functions on by Sobolev embedding theorem.
Let be the subalgebra generated by the rational functions with poles outside . When , the unit disc, we call Sobolev disk algebra. For , the multiplication operator on is defined by , . Then , where is the identity in and is the algebra generated by and identity. In fact, consists of all analytic functions in . We have the following properties of the space and the multiplication operators on it.
Proposition 1.1 (see [1]).
(i) Hilbert space has an orthogonal basis , where
(ii) As a functional Hilbert space, has reproducing kernel which is
Then for ,
(iii) If is analytic on , then if and only if .
(iv) The operator admits the following matrix representation with respect to :
Note that is a subset of the disk algebra , hence a subset of . Because of the special definition of the inner product and the complex behavior of the boundary value, the structure of the space is much more complicated than or . For more details about the Sobolev disk algebra, the reader refers to [1–3].
Let be a complex separable Hilbert space and denote the collection of bounded linear operators on . One of the basic problems in operator theory is to determine when two operators and in are similar. A quantity (quantities) or a property (properties) is similarity invariant (invariants) if has and implies that has [2]. From this point of view, one of the basic problems in operator theory mentioned above is to determine the similarity invariants. There have already been a lot of results on the similarity invariants of operators, especially that of nonadjoint operators, which can be found in, for example, [4–6]. In [7], Wang et al. proved that in , is similar to if and only if is an -Blaschke product. In this paper, we study the similarity invariant of the multiplication operators in , when is univalent analytic on or is strongly irreducible.
It is well known that the commutant of a bounded linear operator or operators on a complex, separable Hilbert space plays an important role in determining the structure of this operator or these operators. The commutant of a multiplication operator on Sobolev disk algebra has been studied in the literature (see [1–3]). In this paper, we describe the commutant of the multiplication operator when is an -Blaschke product. And by this result, we generalize the result which is obtained by Liu and Wang in [3]. Moreover, we study the commutants of the multiplication operators whose symbols are composite functions, univalent analytic functions, or entire functions.
2. Similarity Invariant of Some Multiplication Operators
In this section, we will characterize the similarity invariant of some multiplication operators on Sobolev disk algebra. Here, we briefly recall some background information.
For in , let , , and be the spectrum, point spectrum, and essential spectrum of a bounded linearly operator , respectively. An operator in is said to be a Cowen-Douglas operator with index if there exists , a connected open subset of complex plane , and , a positive integer, such that(i); (ii) for in ;(iii) for in ;(iv), where (iv) is equivalent to (iv)′ ([8]); there exists in , such that .
denotes the collection of Cowen-Douglas operators with index .
For , the set of operators which commute with it is . That is . Operator is strongly irreducible if there is no nontrivial idempotent in [8, 9]. Denote the set of all strongly irreducible operators on .
Definition 2.1. Let be a Hilbert space and be in . is said to be a Rosenblum operator on if for arbitrary .
Lemma 2.2 (see [1]). Let be in , then(i);(ii), where denotes the boundary of the unit disc ;(iii)let and . Denote the component of containing as , then , where is the number of the zeros of in .
Lemma 2.3 (see [10]). Set and . Then .
Theorem 2.4. Let and be in and be univalent and analytic on . Then if and only if .
Proof. “”: Set . By Lemma 2.2, we have
where is the unit circle. Since and are univalent and analytic on , then
Therefore,
“”: Set . Because is univalent analytic from to , is also univalent analytic. Then is injective and surjective analytic function on . If , there exists a Möbius transform with and a complex number with such that (see [11]). Therefore . By Lemma 2.3, .
Lemma 2.5 (see [3]). Given , the following are equivalent:(i);(ii);(iii).
Theorem 2.6. Let and is univalent analytic on . if and only if there exists a function such that , where and .
Proof. “”: Suppose that is not univalent on . There exists some such that the number of zeros of on is . By Lemma 2.2, where is a connected open subset of . Since , we have . This contradicts to that is a strongly irreducible operator (see Lemma 2.5). So is univalent analytic on . By the proof of Theorem 2.4, there exists a function such that , where and .
“”: By the conditions of the theorem, is univalent analytic on . Since , we have by Theorem 2.4.
For any operator on Hilbert space and any integer , let denote the direct sum of copies of on .
Lemma 2.7 (see [2]). Let be strongly irreducible Cowen-Douglas operators. Assume that and , where , are natural numbers. Then for each maximal ideal of , must be one of the following two forms: where is a maximal ideal of , .
Theorem 2.8. Let and . The following statements are equivalent:(i);(ii)there exist and such that where denotes the identity operator of .
Proof. Let . Set be invertible in and . Then and are what we want.
Since and are in , we have and that are strongly irreducible and in by Lemma 2.5. Computations show
Suppose that . By Lemma 2.7, each maximal ideal of must be one of the following two forms
where and are the maximal ideals of and , respectively. We can assume that admits the first form. Then
It follows that
This contradicts that is a maximal ideal. So and .
3. The Commutant Algebra of Multiplication Operator
In [3], Liu and Wang give the following proposition.
Proposition 3.1. Let , , . Then .
Let , () be -Blaschke product. Considering is a special -Blaschke product, we study the commutant of where . The following theorem is obtained, and by this result, the above proposition is generalized.
Theorem 3.2. Let with where does not vanish on . If there exists such that can be divided by each , then .
To prove the above theorem, we need the following lemmas.
Lemma 3.3 (see [7]). Given if and only if is an -Blaschke product.
Lemma 3.4 (see [12]). Let be a nilpotent operator on and let . If satisfy and , then .
Now we will prove Theorem 3.2.
Proof. From , we have . Then, from Lemma 3.3, there exists an invertible operator such that . It follows that
where . So we only need to prove
Since
we have
where is an identity matrix. Computations show that if and only if
Therefore,
Set , and . Then for each , there exists such that . So
where . For , and pairwise commute. Hence, pairwise commute too. Since and
for all ,
Therefore, is nilpotent operator. Now we set , that is . So
So we only need to prove that . In fact, if ,
It follows from that . Namely, .
We are now in need to prove that . Suppose that . Since does not vanish on , and so . Because ,
It follows from that and are both the invariant subspaces of . Since
admits the matrix representation (3.5) with the above decomposition. So
From , we have
Comparing the entries of both sides, we have by Lemma 3.4. Comparing the entries of and , we have
It follows from Lemma 3.4 that
Setting , we have
Inductively, if , where , we need to prove . Comparing the entries of and , we have
Therefore, by Lemma 3.4,
Computations show . Since is the form of (3.5), . So .
Corollary 3.5. Let with where does not vanish on . If there exists that can be divided by , then .
By the following lemma, we discuss the commutant of the multiplication operators whose symbols are composite functions in .
Lemma 3.6. For in and in the following are equivalent:(i);(ii)for all , ;(iii)there is a set such that and for all , .
Proof. Let . For all and we have
Let .
Let with for all . For and , we have
Since is not a Blaschke sequence, this means . Therefore .
Lemma 3.7 (see [13]). Suppose is a surjective analytic function and for each , is the number of points in . Then
Proposition 3.8. Let be in and is analytic on . Suppose for each , there are points in . Then for , we have .
Proof. By the Embedding Theory of Sobolev space, . Therefore
By Lemma 3.7,
Since is analytic on , , is bounded on . Hence
Therefore,
By (3.24), (3.25), and (3.27), we have .
For all , because , . For all
so that
Set . By Lemma 3.6, for all , . Hence, and we have .
Corollary 3.9. If , and , .
Proposition 3.10. Let be in and is an -Blaschke product. If , then .
Proof. From [3], we know that . By Lemma 3.3, . Then there exists an invertible operator in such that . Since , we have . Therefore, we will only prove that . Set and Since we have for . So and . Similarly, . Hence .
Let be an injective function in and . Then for each , it is obvious that is not in . Denote the component of containing as , then is the only zero point of in . By Lemma 2.2, is a Cowen-Douglas operator with index 1. By Lemma 2.5, we have . So the following corollary is obtained.
Corollary 3.11. Let be a univalent analytic function in and be an -Blaschke product. Then .
Lemma 3.6 shows if is in and is in , then Easy examples show that is not true. The following proposition shows that if , .
Lemma 3.12 (see [1]). Let , and where are pairwise distinct, . Choose such that Then there exists a linearly independent set such that .
Let and be the th root of 1, that is, and . Let denote the Vandermonde determinant of order : For , , the -cofactor will be denoted by .
Lemma 3.13 (see [1]). if and only if for all and , where , for some in .
Proposition 3.14. For all , .
Proof. By Lemma 3.6, . Now we prove that .
Since
by Lemma 3.12,
Set with . For all , we define an operator as follows:
By Lemma 3.13, . For all ,
For ,
Therefore,
that is . Then and we have .
Easy examples show that, in general, the converse of Proposition 3.8 is false. But the following case is an exception. To prove it, we need the following lemma.
Lemma 3.15. Let be in and , analytic on . Then is in .
Proof. Since is analytic on , we have , hence, From being in , we have Because is monotonically decreasing, for all . So and this shows that is in .
Proposition 3.16. If and , then there exists being in such that .
Proof. By Proposition 3.14, for all , we have
For each , we can find such that . We define on by and is well defined. Indeed, set . Then
Hence .
For , we have . Therefore,
If , . Since
there exists such that . Hence,
So is analytic on . By Lemma 3.15, we have and .
For each and , denote the winding number of at . Define
Proposition 3.17. If is a nonconstant entire function and , then .
Proof. By Theorem 1 in [14], there exists an entire function such that and . Since is an entire function, , , and are all bounded and analytic on . So . By , there is only one zero of in for some . By Lemma 2.2, . By Proposition 3.10, .
Acknowledgment
The research is supported by NSFC 11001078.