-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.
The structure of -algebras generated by isometry is determined in [1, 2]. The structure is the same with the structure of -algebras generated by unilateral weighted shift operators, that is, the structure of -algebras generated by multiplication operators with the independent variable in the Hardy space on the unit disc. The analogue of the unit disc on is the polydisc or the unit ball The structures of -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 -algebras generated by system of unilateral weighted shifts.
Let be a multiindex of integers and denotes for the multi-index . Here is another multi-index , where is the Kronocker symbol.
Let be an orthonormal basis of a separable complex Hilbert space and let be a bounded net of complex numbers. Denote by the bounded linear operators whose effect on the elements of basis of is given as . A family of operators, denoted by , is called a system of unilateral weighted shifts, and the numbers of are called the weights of the system. It is known from [3, page 209, Corollary 2] that for shifts with nonzero weights , without loss of generality we may always assume the weights are a set of positive real numbers, that is, the system positive. It is possible to show that if there exists a solution for the multivariable moment problem for the net where , , that is, if there exists the probability measure on ( times) such that then the system is unitarily equivalent to the system of multiplication operators by the independent variables , , on the space where , .
The reader can find for more details of such operators in the article by Jewell and Lubin  and Ergezen and Sadik . Furthermore the papers of Curto and Yoon  and Curto and Yan  are closely related to our study.
2. -Algebras Generated by a System of Unilateral Weighted Shifts
Let denote the family of the systems which satisfy the functional model defined above. Moreover, let be a subset of defined by
Theorem 2.1 (see [4, page 25, Theorem 2]). Let . A necessary and sufficient condition for the operator algebra generated by the system to be isometrically isomorphic to the polydisc algebra is that belongs to .
This theorem will be helpful in studying the structure of -algebra generated by .
Let denote the orthogonal projection of onto and let lie in . Then the Toeplitz operator for in .
Without loss of generality we may take . The following theorems for were given by Sadikov .
Theorem 2.2. Let . If the algebra generated by the system is polydisc algebra then the commutator ideal of contains properly the ideal of compact operators and the quotient space is isometrically isomorphic to and , where is unit circle and is the two point space.
Corollary 2.3. Let . Then necessary and sufficient condition for to be Fredholm is that is nonvanishing for and is homotopic to constant.
Theorem 2.2 and Corollary 2.3 are proved by using the methods of Douglas and Howe in their study  and Curto and Muhly in .
Let and let be subset of defined by
Theorem 2.4 (see [10, p. 1932, Theorem 2]). Let . A necessary and sufficient condition for the operator algebra generated by the system to be isometrically isometric to the ball algebra is that belongs to .
Theorem 2.5. Let . If the algebra generated by the system is ball algebra then contains ideal of compact operators and . The quotient is naturally identified with by a map .
It is enough to show compactness of the operators , for proving that commutant of the algebra is . For this, we just study the case . Then the case is similarly showed, as well. With the basic computations, we obtain . If we show converges the zero when then the proof is completed. For this, we need the following lemma.
Lemma 2.6. Let be a measure determined by the system and . then the expression converges to zero when .
In view of the following process, the lemma is proved. Without loss of generality, we can take . Hence the second fractional of the expression becomes . It is enough to show goes to zero when . Take and for given . Consider and . We have , where . It easily shows that and for all . Moreover, take ; then we have for all , where . Hence, there exists such that for all it is obtained .
It follows from Lemma 2.6 and a well-known result in -algebras [2, page 212, Proposition 1] that .
3. An Application
Throughout this section, we follow the notations and definitions in the preceding section.
Let be separable complex Hilbert space, let denote the algebra of linear bounded operators on and let be identity operator. Consider an operator where and are the elements of a subalgebra of such that the image is a commutative subalgebra of the algebra under the natural quotient map from to and denotes an automorphism in the algebra , that is, , where and belong to . It is obvious that if then , where and .
Theorem 3.1 (see ). If the operator has an inverse in then is Fredholm operator.
Using the Theorem 3.1 we take and consider the operator , where and are Toeplitz operators in and the operators and satisfy the conditions given above.
Moreover if we take into account the orthonormal projection has the form where is the reproducing Bergman kernel of the functional space , then the equation is written in the form
Hence we have the following theorem.
Theorem 3.2. If the function does not vanish in , then all of Noether’s theorems is true for (3.3). In particular, if we take , and then (3.3) has the form where is the surface measure in .
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