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

Abstract

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

1. 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 polydiscΔ^𝑁=𝑧=𝑧₁,𝑧₂,…,𝑧_𝑁∈ℂ^𝑁∶𝑧_𝑖<1,1≤𝑖≤𝑁(1.1) or the unit ball 𝐵^𝑁=𝑧=𝑧₁,𝑧₂,…,𝑧_𝑁∈ℂ^𝑁∶𝑧₁²+𝑧₂²+⋯+𝑧_𝑁²<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.

Let 𝐼 be a multiindex (𝑖₁,…,𝑖_𝑁) of integers and 𝐼±𝜀_𝑗 denotes (𝑖₁,…,𝑖_𝑗±1,…,𝑖_𝑁) for the multi-index 𝐼. Here 𝜀_𝑗 is another multi-index (𝛿_1𝑗,…,𝛿_𝑁𝑗), where 𝛿_𝑖𝑗is 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_𝑗𝑒_𝐼=𝜔_𝐼,𝑗𝑒_{𝐼+𝜀𝑗},1≤𝑗≤𝑁. A family of 𝑁 operators, denoted by 𝐴=(𝐴₁,𝐴₂,…,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 {𝛽²_𝐼}_𝐼≥0 where 𝛽_{𝐼+𝜀𝑗}=𝜔_𝐼,𝑗𝛽_𝐼, 𝛽₀=1, that is, if there exists the probability measure 𝜈_𝐴 on [0,1]^𝑁=[0,1]×⋯×[0,1](𝑁 times) such that 𝛽²_𝐼=_[0,1]𝑁𝑟^2𝑖1₁𝑟^2𝑖2₂⋯𝑟^2𝑖𝑁_𝑁𝑑𝜈𝑟₁,𝑟₂,…,𝑟_𝑁,(1.3) 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 𝐻²Δ^𝑁,𝜇_𝐴=𝑓=_𝐼≥0𝑓_𝐼𝑧^𝐼∶𝑧∈Δ^𝑁,_𝐼≥0𝑓_𝐼²𝛽²_𝐼<∞,(1.4) where 𝜇_𝐴=𝜈_𝐴𝑑𝜃₁𝑑𝜃₂⋯𝑑𝜃_𝑁, 0≤𝜃_𝑖≤2𝜋.

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.

2. 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 Ω₁ be a subset of Ω defined byΩ₁=𝐴∈Ω∶𝜈_𝐴(𝑈(1,1,…,1))>0forarbitraryneighborhood𝑈(1,1,…,1)ofthepoint(1,1,…,1)∈[0,1]^𝑁.(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 Ω₁.

This theorem will be helpful in studying the structure of 𝐶^∗-algebra 𝐶^∗(𝐴) generated by 𝐴∈Ω₁.

Let 𝑃 denote the orthogonal projection of 𝐿₂(Δ^𝑁,𝜇_𝐴) onto 𝐻²(Δ^𝑁,𝜇_𝐴) and let 𝜓 lie in 𝐶(supp𝜇_𝐴). Then the Toeplitz operator 𝑇_𝜓𝑓=𝑃(𝜓𝑓) for 𝑓 in 𝐻²(Δ^𝑁,𝜇_𝐴).

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∈𝑇² and 𝜓|_𝑇²is homotopic to constant.

Theorem 2.2 and Corollary 2.3 are proved by using the methods of Douglas and Howe in their study [8] and Curto and Muhly in [9].

Let 𝑆₁=𝑟₁,𝑟₂∈[0,1]×[0,1]∶𝑟²₁+𝑟²₂,𝑆≤1₁=𝑟₁,𝑟₂∈[0,1]×[0,1]∶𝑟²₁+𝑟²₂=1,(2.2) and let Ω₂ be subset of Ω defined by Ω₂=𝐴∈Ω∶𝜈_𝐴(𝑈(𝑎))>0forarbitraryneighborhood𝑈(𝑎)ofthearbitrarypoint𝑎∈𝑆₁.(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 Ω₂.

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(supp𝜇_𝐴),𝐾∈𝒦}. The quotient 𝐶^∗(𝐴)/𝒦 is naturally identified with 𝐶(𝑆₃) by a map 𝜎(𝑇_𝜓+𝐾)=𝜓|_𝑆3.

It is enough to show compactness of the operators 𝐴^∗_𝑖𝐴_𝑖−𝐴_𝑖𝐴^∗_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^∗₂𝐴₂−𝐴₂𝐴^∗₂)𝑒_(𝑚,𝑛)=(𝜔²_{(𝑚,𝑛),2}−𝜔²_{(𝑚,𝑛−1),2})𝑒_(𝑚,𝑛). If we show 𝜔²_{(𝑚,𝑛),2}−𝜔²_{(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∈Ω₂ and 𝐼=(𝑚,𝑛). then the expression ∫_𝑆1𝑟₁^2𝑚𝑟₂^2𝑛+2𝑑𝜈𝑟₁,𝑟₂∫_𝑆1𝑟₁^2𝑚𝑟₂^2𝑛𝑑𝜈𝑟₁,𝑟₂−∫_𝑆1𝑟₁^2𝑚𝑟₂^2𝑛𝑑𝜈𝑟₁,𝑟₂∫_𝑆1𝑟₁^2𝑚𝑟₂^2𝑛−2𝑑𝜈𝑟₁,𝑟₂(∗) converges to zero when |𝐼|→∞.

In view of the following process, the lemma is proved. Without loss of generality, we can take 𝑛=1. Hence the second fractional of the expression (∗) becomes 𝐼₁/𝐼₂∶=∫_𝑆1𝑟₁^2𝑚𝑟²₂𝑑𝜈(𝑟₁,𝑟₂)/∫_𝑆1𝑟₁^2𝑚𝑑𝜈(𝑟₁,𝑟₂). It is enough to show 𝐼₁/𝐼₂ goes to zero when 𝑚→∞. Take 𝑟₂₀=𝜀/2 and 𝑟₁₀=1−(𝜀/2)=1−𝑟²₂₀ for given 𝜀>0. Consider 𝑆₁₀∶={(𝑟₁,𝑟₂)∶𝑟₁₀<𝑟₁≤1}∩𝑆₁ and 𝑆₁₁∶=𝑆₁/𝑆₁₀. We have 𝐼₁=𝐼₁₀+𝐼₁₁, where 𝐼₁₀=∫_𝑆10𝑟₁^2𝑚𝑟²₂𝑑𝜈(𝑟₁,𝑟₂). It easily shows that 𝐼₁₀≤(𝜀/2)𝐼₂ and 𝐼₁₁≤𝑟^2𝑚₁₀𝜈_𝐴(𝑆₁) for all 𝑚. Moreover, take 𝑟₁₁=(1+𝑟₁₀)/2; then we have 𝐼₂≥𝑟^2𝑚₁₁𝜈_𝐴(𝑆₁₀₁) for all 𝑚, where 𝑆₁₀₁={(𝑟₁,𝑟₂)∶𝑟₁₁<𝑟₁<1}∩𝑆₁. Hence, there exists 𝑀>0 such that for all 𝑚>𝑀 it is obtained 𝐼₁/𝐼₂<𝜀.

It follows from Lemma 2.6 and a well-known result in 𝐶^∗-algebras [2, page 212, Proposition 1] that 𝒦⊂𝐶^∗(𝐴).

3. An Application

Throughout this section, we follow the notations and definitions in the preceding section.

Let 𝐻 be separable complex Hilbert space, let𝐵(𝐻) denote the algebra of linear bounded operators on 𝐻 and let 𝐼 be identity operator. Consider an operator𝑇=𝐵₁+𝐵₂𝑆+𝐾,(3.1) where 𝐵₁ and 𝐵₂ 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(𝐵₁)𝜏(𝐵^′₁)−𝜏(𝐵₂)𝜏(𝐵^′₂) has an inverse in 𝜏(Λ) then 𝑇=𝐵₁+𝐵₂𝑆+𝐾 is Fredholm operator.

Using the Theorem 3.1 we take 𝐴∈Ω₂ and consider the operator 𝑇_𝜓1+𝑇_𝜓2𝑆+𝐾, where 𝑇_𝜓1 and 𝑇_𝜓2 are Toeplitz operators in 𝐶^∗(𝐴) and the operators 𝑆 and 𝑇 satisfy the conditions given above.

Moreover if we take into account the orthonormal projection 𝑃 has the form 𝑃𝑓𝑧₁,𝑧₂=_Δ2𝐾(𝑧,𝑙)𝑓𝑙₁,𝑙₂𝑑𝜇_𝐴𝑙₁,𝑙₂,(3.2) where 𝐾 is the reproducing Bergman kernel of the functional space 𝐻²(Δ²,𝜇_𝐴), then the equation 𝑇𝑓=𝜑 is written in the form _Δ2𝐾(𝑧,𝑙)𝜓₁𝑙₁,𝑙₂𝑓𝑙₁,𝑙₂𝑑𝜇_𝐴𝑙₁,𝑙₂+_Δ2𝐾(𝑧,𝑙)𝜓₂𝑙₁,𝑙₂𝑓𝑙₁,𝑙₂𝑑𝜇_𝐴𝑙₁,𝑙₂+𝐾𝑓𝑧₁,𝑧₂=𝜑𝑧₁,𝑧₂.(3.3)

Hence we have the following theorem.

Theorem 3.2. If the function 𝜓₁(𝑧₁,𝑧₂)𝜓₁(𝑧₂,𝑧₁)−𝜓₂(𝑧₁,𝑧₂)𝜓₂(𝑧₂,𝑧₁) does not vanish in 𝑆₃, then all of Noether’s theorems is true for (3.3). In particular, if we take 𝜔_𝐼,1=(𝑚+1)/(2+𝑚+𝑛), 𝜔_𝐼,2=(𝑛+1)/(2+𝑚+𝑛) and 𝑆𝑓(𝑙₁,𝑙₂)=𝑓(𝑙₂,𝑙₁) then (3.3) has the form _𝑆3𝜓₁𝑙₁,𝑙₂𝑓𝑙₁,𝑙₂1−𝑧₁𝑒₁−𝑧₂𝑒₂²𝑑𝑠+_𝑆3𝜓₂𝑙₁,𝑙₂𝑓𝑙₂,𝑙₁1−𝑧₁𝑒₁−𝑧₂𝑒₂²𝑑𝑠+𝐾𝑓𝑧₁,𝑧₂=𝜑𝑧₁,𝑧₂,(3.4) where 𝑑𝑠 is the surface measure in 𝑆₃.