C ∗-Algebras Generated by a System of Unilateral Weighted Shifts and Their Application

We study the structure -algebras generated by a system of unilateral weighted shifts. Finally the obtained results are applied to a class of integral equations.


Introduction
The structure of C * -algebras generated by isometry is determined in 1, 2 . The structure is the same with the structure of C * -algebras generated by unilateral weighted shift operators, that is, the structure of C * -algebras generated by multiplication operators with the independent variable in the Hardy space on the unit disc. The analogue of the unit disc on C N is the polydisc or the unit ball B N z z 1 , z 2 , . . . , z N ∈ C N : |z 1 | 2 |z 2 | 2 · · · |z N | 2 < 1 .

1.2
The structures of C * -algebras generated by multiplication operators with the independent variable in the Hardy space on the unit ball and polydisc are different. To understand this difference we study the structure of C * -algebras generated by system of unilateral weighted shifts. 2

Journal of Function Spaces and Applications
Let I be a multiindex i 1 , . . . , i N of integers and I ± ε j denotes i 1 , . . . , i j ± 1, . . . , i N for the multi-index I. Here ε j is another multi-index δ 1j , . . . , δ Nj , where δ ij is the Kronocker symbol.
Let {e I } I≥0 be an orthonormal basis of a separable complex Hilbert space H and let {ω I,j : I ≥ 0, 1 ≤ j ≤ N} be a bounded net of complex numbers. Denote by A j the bounded linear operators whose effect on the elements of basis {e I } I≥0 of H is given as A j e I ω I,j e I ε j , 1 ≤ j ≤ N. A family of N operators, denoted by A A 1 , A 2 , . . . , A N , is called a system of unilateral weighted shifts, and the numbers of {ω I,j : I ≥ 0, 1 ≤ j ≤ N} are called the weights of the system. It is known from 3, page 209, Corollary 2 that for shifts with nonzero weights {ω I,j }, without loss of generality we may always assume the weights are a set of positive real numbers, that is, the system A positive. It is possible to show that if there exists a solution for the multivariable moment problem for the net then the system A is unitarily equivalent to the system of multiplication operators by the independent variables z j , 1 ≤ j ≤ N, on the space The reader can find for more details of such operators in the article by Jewell and Lubin 3 and Ergezen and Sadik 4 . Furthermore the papers of Curto and Yoon 5 and Curto and Yan 6 are closely related to our study.

C * -Algebras Generated by a System of Unilateral Weighted Shifts
Let Ω denote the family of the systems A which satisfy the functional model defined above. Moreover, let Ω 1 be a subset of Ω defined by

2.1
Theorem 2.1 see 4, page 25, Theorem 2 . Let A ∈ Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isomorphic to the polydisc algebra is that A belongs to Ω 1 .
This theorem will be helpful in studying the structure of C * -algebra C * A generated by A ∈ Ω 1 .
Let P denote the orthogonal projection of L 2 Δ N , μ A onto H 2 Δ N , μ A and let ψ lie in C supp μ A . Then the Toeplitz operator T ψ f P ψf for f in H 2 Δ N , μ A .

Journal of Function Spaces and Applications 3
Without loss of generality we may take N 2. The following theorems for N 1 were given by Sadikov 7 . Theorem 2.2. Let A ∈ Ω. If the algebra generated by the system A is polydisc algebra then the commutator ideal J of C * A contains properly the ideal of compact operators K and the quotient space J/K is isometrically isomorphic to C T × {0, 1} ⊕ K and C * A /J C T × T , where T is unit circle and {0, 1} is the two point space. Corollary 2.3. Let T ψ ∈ C * A . Then necessary and sufficient condition for T ψ to be Fredholm is that ψ z is nonvanishing for z ∈ T 2 and ψ| T 2 is homotopic to constant.

2.3
Theorem 2.4 see 10, p. 1932, Theorem 2 . Let A ∈ Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isometric to the ball algebra is that A belongs to Ω 2 .
Theorem 2.5. Let A ∈ Ω. If the algebra generated by the system A is ball algebra then C * A contains K ideal of compact operators and C * A {T ψ K : ψ ∈ C supp μ A , K ∈ K}. The quotient C * A /K is naturally identified with C S 3 by a map σ T ψ K ψ| S 3 .
It is enough to show compactness of the operators A * i A i −A i A * i , i 1, 2 for proving that commutant of the algebra C * A is K. For this, we just study the case i 2. Then the case i 1 is similarly showed, as well. With the basic computations, we obtain A * 2 A 2 − A 2 A * 2 e m,n ω 2 m,n ,2 − ω 2 m,n−1 ,2 e m,n . If we show ω 2 m,n ,2 − ω 2 m,n−1 ,2 converges the zero when |I| → ∞ then the proof is completed. For this, we need the following lemma. Lemma 2.6. Let ν A be a measure determined by the system A ∈ Ω 2 and I m, n . then the expression S 1 r 2m 1 r 2n 2 2 dν r 1 , r 2 converges to zero when |I| → ∞.
It follows from Lemma 2.6 and a well-known result in C * -algebras 2, page 212, Proposition 1 that K ⊂ C * A .

An Application
Throughout this section, we follow the notations and definitions in the preceding section.
Let H be separable complex Hilbert space, let B H denote the algebra of linear bounded operators on H and let I be identity operator. Consider an operator where B 1 and B 2 are the elements of a subalgebra Λ of B H such that the image τ Λ is a commutative subalgebra of the algebra B H /K under the natural quotient map τ from B H to B H /K and S denotes an automorphism in the algebra τ Λ , that is, τ S τ B τ S −1 τ B , where B and B belong to Λ. It is obvious that if B ∈ Λ then SBS −1 B K, where B ∈ Λ and K ∈ K. Theorem 3.1 see 11 . If the operator τ B 1 τ B 1 − τ B 2 τ B 2 has an inverse in τ Λ then T B 1 B 2 S K is Fredholm operator.