International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 378217, 17 pages

doi:10.1155/2012/378217

Research Article

A Similarity Invariant and the Commutant of some Multiplication Operators on the Sobolev Disk Algebra

Ruifang Zhao

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 31 March 2012; Accepted 19 July 2012

Academic Editor: Gideon Schechtman

Copyright © 2012 Ruifang Zhao. 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.

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.

References

Z. Wang and Y. Liu, “Multiplication operators on Sobolev disk algebra,” Science in China Series A, vol. 48, no. 10, pp. 1395–1410, 2005.

C. L. Jiang and Z. Y. Wang, Structure of Hilbert Spaces Operators, World Scientific, Singapore, 2006.

Y. Liu and Z. Wang, “The commutant of the multiplication operators on Sobolev disk algebra,” Journal of Analysis and Applications, vol. 2, no. 2, pp. 65–86, 2004.

D. A. Herrero, “Spectral pictures of operators in the Cowen-Douglas class and its closure,” Journal of Operator Theory, vol. 18, no. 2, pp. 213–222, 1987.

D. A. Herrero, Approximation of Hilbert Space Operatrs I, Research Notes in Mathematics 224, Longman, Essex, UK, 2nd edition, 1989.

J. B. Conway, Course in Functional Analysis, vol. 96 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1990.

Z. Wang, R. Zhao, and Y. Jin, “Finite Blaschke product and the multiplication operators on Sobolev disk algebra,” Science in China Series A, vol. 52, no. 1, pp. 142–146, 2009.

C. Jiang and Z. Wang, Strongly Irreducible Operators on Hilbert Space, vol. 389 of Pitman Research Notes in Mathematics Series, Longman, Harlow, UK, 1998.

F. Gilfeather, “Strong reducibility of operators,” Indiana University Mathematics Journal, vol. 22, pp. 393–397, 1972-1973.

Y.-Q. Liu and Z.-Y. Wang, “Invariant subspaces of Sobolev disk algebra,” Journal of Mathematical Research and Exposition, vol. 26, no. 2, pp. 233–238, 2006.

J. B. Conway, Functions of One Complex Variable I, vol. 11 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1978.

J. A. Deddens and T. K. Wong, “The commutant of analytic Toeplitz operators,” Transactions of the American Mathematical Society, vol. 184, pp. 261–273, 1973.

J. B. Conway, Functions of One Complex Variable. II, vol. 159 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1995.

I. N. Baker, J. A. Deddens, and J. L. Ullman, “A theorem on entire functions with applications to Toeplitz operators,” Duke Mathematical Journal, vol. 41, pp. 739–745, 1974.