The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.
Citation: Xiaojun Xie, Saratha Sathasivam, Hong Ma. Modeling of 3 SAT discrete Hopfield neural network optimization using genetic algorithm optimized K-modes clustering[J]. AIMS Mathematics, 2024, 9(10): 28100-28129. doi: 10.3934/math.20241363
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The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.
As usual, let H denote a complex Hilbert space and B(H) the set of all bounded linear operators on H. Let ˉz be the usual conjugation of complex number z, that is, if z=x+iy, then ˉz=x−iy. Let T∗ denote the adjoint operator of T for T∈B(H).
We continuously introduce some notations. For a complex number z, let ℜz denote the real part of z and ℑz the imaginary part of z. Let C denote the complex plane, Π the right half-plane {z∈C:ℜz>0}, and A2α(Π) the weighted Bergman space on Π.
We first need the following definition.
Definition 1.1. An operator C:H→H is said to be a conjugation if it satisfies the following conditions:
(i) conjugate-linear: C(αx+βy)=ˉαC(x)+ˉβC(y), for α,β∈C and x,y∈H;
(ii) isometric: ||C(x)||=||x||, for all x∈H;
(iii) involutive: C2=Id, where Id is an identity operator.
One can see [7] for more information about conjugations. Actually, there exists many conjugations on A2α(Π) such as Jf(z)=¯f(ˉz).
Definition 1.2. Let C:H→H be a conjugation and T∈B(H). If CTC=T∗, then T is said to be complex symmetric with C.
Complex symmetric operators on abstract Hilbert space were studied by Garcia, Putinar, and Wogen in [7,8,9,10]. Afterwards, one started to consider such operators on function spaces. For example, Noor et al. in [24] characterized the complex symmetric composition operators on Hardy space of the right half-plane. See also [5,11,12,13,18,19,22,23,30,33] for the studies of such operators on function spaces. After a long time of research, people find that many operators are complex symmetric operators such as normal operators, Hankel operators, and Volterra integration operators. Complex symmetric operators have been extensively used in theoretic and application aspects (see [6]).
Ptak et al. in [26] introduced an interesting class of operators named complex normal operators and proved that the class of the complex normal operators properly contains complex symmetric operators.
Definition 1.3. Let C be a conjugation on H and T∈B(H). If C(T∗T)=(TT∗)C, then T is said to be complex normal with C.
In the recent paper [31], Wang et al. studied the structure of complex normal operators and provided a refined polar decomposition of complex normal operators. Recently, Bhuia in [1] studied complex normal weighted composition operators on Fock space and provided some properties of complex normal weighted composition operators. A direct proof shows that complex symmetric operators are always complex normal. Thus, complex normality can be viewed as a generalized complex symmetry. Also, in [26], basic properties of complex normal operators are developed. In particular, special attention is paid to complex normal operators on finite dimensional spaces, L2 type spaces, and the Hardy space H2.
Because of the above-mentioned studies, we can try to study the complex symmetric or complex normal weighted composition operators on other analytic function spaces. Coincidentally, when we are considering such problems, we find that Hai et al. in [14] studied the following conjugations on A2α(Π)
Jg(z)=¯g(ˉz),Jsg(z)=¯g(ˉz+is),J∗f(z)=1zα+2¯g(1ˉz), | (1.1) |
where s∈R. Since it is difficult to give a proper description of the adjoint for the weighted composition operators with the general symbols on A2α(Π), in [14] they just characterized the complex symmetric weighted composition operators with the symbols in (Ⅰ)–(Ⅲ) on A2α(Π). These symbols are defined as follows:
(Ⅰ)
τ(z)=1(z−c)α+2,ϕ(z)=−a−bz−c, |
where coefficients satisfy
{either ℜa=ℑb=0,ℜb<0,ℜc≤0,or ℜa<0≤−ℜc+ℜb+|b|2ℜa. | (1.2) |
(Ⅱ)
τ(z)=δ(z+μ+iη)α+2,ϕ(z)=μ, |
where coefficients satisfy δ∈C, μ∈Π, and η∈R.
(Ⅲ)
τ(z)=λ,ϕ(z)=z+γ, |
where coefficients satisfy λ∈C and γ∈Π.
Next, we will provide the research motivations of this paper. With the basic questions such as boundedness and compactness settled, more attention has been paid to the study of the topological structure of the composition operators or weighted composition operators in the operator norm topology. In this research background, Shapiro and Sundberg in [27] posed a question on whether two composition operators belong to the same connected component, when their difference is compact. In the study of difference of composition operators, some interesting phenomena were found. For example, there is no compact composition operators on weighted Bergman space on the half-plane (see [20]), but there is compact difference of composition operators on this space (see [3]); two noncompact composition operators can induce compact difference of composition operators on weighted Bergman space on the unit disk (see [21]). Perhaps due to these interesting phenomena, people initiated the study of difference of composition operators or weighted composition operators, which has become a very active topic (see [16,21,28]).
Motivated by the above-mentioned studies, a natural problem is to characterize complex symmetric difference of composition operators or weighted composition operators on analytic function spaces. To this end, we try to consider this problem on weighted Bergman space A2α(Π) by using the weighted composition operators with symbols in (Ⅰ)–(Ⅲ). As we expected, we find that the difference of such weighted composition operators is complex symmetric on weighted Bergman space A2α(Π) with the conjugations in (1.1) if and only if each weighted composition operator is complex symmetric. This is an interesting phenomenon, but it may be not right for the general case, that is, from the complex symmetry of the operator T=T1+T2, where T1,T2∈B(H), it cannot deduce the complex symmetries of the operators T1 and T2. On the other hand, it is well known that there is no compact composition operators on the weighted Bergman space A2α(Π). Maybe for that reason, there isarded as an useful supplement of the weighted composition operators on A2α(Π).
Throughout the paper, we always assume that α is a nonnegative integer, since for any w,z∈C and α>0, (wz)α≠wαzα while the equality holds when α is a nonnegative integer.
Let H(Π) be the set of all analytic functions on Π, dA be the area measure on Π, and dAα(z)=2α(α+1)π(ℜz)αdA(z). The weighted Bergman space A2α(Π) consists of all f∈H(Π) such that
‖f‖2A2α(Π)=∫Π|f(z)|2dAα(z)<∞. |
Moreover, this norm is induced by the inner product
⟨f,g⟩A2α(Π)=∫Πf(z)¯g(z)dAα(z). |
A2α(Π) is a Hilbert space with this inner product, and the reproducing kernel is
Kαw(z)=2α(α+1)(z+¯w)α+2,z∈Π. |
That is,
f(z)=⟨f,Kαz⟩A2α(Π)=∫Πf(w)¯Kαz(w)dAα(w) |
for any f∈A2α(Π) and z∈Π. One can see [4] for more information on A2α(Π).
Let φ be an analytic self-mapping of Π and u∈H(Π). The weighted composition operator induced by the symbols u and φ on or between some subspaces of H(Π) is defined by
Wu,φf(z)=u(z)f(φ(z)). |
From the definition, it follows that when u≡1, Wu,φ is the composition operator, denoted by Cφ; when φ(z)=z, Wu,φ is the multiplication operator, denoted by Mu.
It is an interesting topic to provide the characterizations of the symbols u and φ which induce bounded or compact weighted composition operators. Recently, several authors have studied the composition operators and weighted composition operators on weighted Bergman space of the half-plane. For example, Elliott et al. in [4] characterized the bounded composition operators and proved that no composition operator on the weighted Bergman space of the upper half-plane is compact. Sharma et al. in [29] characterized the bounded weighted composition operators on vector-valued weighted Bergman spaces of the upper half-plane. Readers can also find some relevant studies about the operators on the weighted Bergman spaces of the upper half-plane and we will not repeat them anymore.
The following result can be directly obtained by utilizing the denseness of the linear span of the functions {Kαw:w∈Π} in A2α(Π).
Lemma 3.1. Let T be a bounded operator on A2α(Π). Then, T is complex symmetric on A2α(Π) with the conjugation C if and only if
(CT−T∗C)Kαw(z)=0 | (3.1) |
for all w,z∈Π.
To study the difference of the operators Wτ,ϕ with the symbols in (Ⅰ)–(Ⅲ) on A2α(Π), we need the following result, where (a) was proved in [32].
Lemma 3.2. (a) Let τ(z)=1(z−c)α+2 and ϕ(z)=−a−bz−c be the symbols defined in (I). Then, on A2α(Π) the following holds
W∗τ,ϕ=W1(z−ˉa)α+2,ˉaˉc−ˉb−ˉczz−ˉa. |
(b) Let τ(z)=δ(z+μ+iη)α+2 and ϕ(z)=μ be the symbols defined in (II). Then, on A2α(Π) the following holds
W∗τ,ϕ=ˉδW1(z+ˉμ)α+2,ˉμ−iη. |
(c) Let τ(z)=λ and ϕ(z)=z+γ be the symbols defined in (III). Then, on A2α(Π) the following holds
W∗τ,ϕ=ˉλCz+ˉγ. |
Proof. (b). From Lemma 3.2 in [32], we have
W∗τ,ϕ=Wˉδ(z+ˉμ)α+2,ˉμ−iη=ˉδW1(z+ˉμ)α+2,ˉμ−iη. |
(c). The proof can be similarly obtained, so we do not provide proof anymore.
First, we characterize the complex symmetric difference of the operator Wτ,ϕ with the symbols in (Ⅰ) on A2α(Π) with the conjugation J.
Theorem 3.1. Let τj(z)=1(z−cj)α+2 and ϕj(z)=−aj−bjz−cj be the symbols in (I) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if a1=c1 and a2=c2.
Proof. For all w,z∈Π, from Lemma 3.2 (a) the following equalities hold
J(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J(1(z−c1)α+22α(α+1)(−a1−b1z−c1+¯w)α+2−1(z−c2)α+22α(α+1)(−a2−b2z−c2+¯w)α+2)=J(2α(α+1)[(¯w−a1)z+a1c1−b1−c1¯w]α+2−2α(α+1)[(¯w−a2)z+a2c2−b2−c2¯w]α+2)=2α(α+1)[(w−ˉa1)z+ˉa1ˉc1−ˉb1−ˉc1w]α+2−2α(α+1)[(w−ˉa2)z+ˉa2ˉc2−ˉb2−ˉc2w]α+2 | (3.2) |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗JKαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(z+w)α+2)=(W1(z−ˉa1)α+2,ˉa1ˉc1−ˉb1−ˉc1zz−ˉa1−W1(z−ˉa2)α+2,ˉa2ˉc2−ˉb2−ˉc2zz−ˉa2)(2α(α+1)(z+w)α+2)=2α(α+1)[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+2−2α(α+1)[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+2. | (3.3) |
Hence, from (3.2), (3.3) and Lemma 3.1, it follows that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if
1[(w−ˉa1)z−ˉc1w+ˉa1ˉc1−ˉb1]α+2−1[(w−ˉa2)z−ˉc2w+ˉa2ˉc2−ˉb2]α+2=1[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+2−1[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+2 | (3.4) |
for all w,z∈Π.
Assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J. Then, from (3.4) we obtain
1[(w−ˉa1)z−ˉc1w+ˉa1ˉc1−ˉb1]α+2−1[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+2=1[(w−ˉa2)z−ˉc2w+ˉa2ˉc2−ˉb2]α+2−1[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+2 | (3.5) |
for all w,z∈Π. From the formula
xn−yn=(x−y)(xn−1+xn−2y+⋯+xyn−2+yn−1), |
we have
(ˉa1−ˉc1)(z−w){[w−ˉa1)z−ˉc1w+ˉa1ˉc1−ˉb1]α+1+⋯+[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+1}[(w−ˉa1)z−ˉc1w+ˉa1ˉc1−ˉb1]α+2[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+2=(ˉa2−ˉc2)(z−w){[w−ˉa2)z−ˉc2w+ˉa2ˉc2−ˉb2]α+1+⋯+[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+1}[(w−ˉa2)z−ˉc2w+ˉa2ˉc2−ˉb2]α+2[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+2 | (3.6) |
for all w,z∈Π. Clearly, if a2≠c2, then from (3.6) we have
ˉa1−ˉc1ˉa2−ˉc2≡[(w−ˉa2)z−ˉc2w+ˉa2ˉc2−ˉb2]α+1+⋯+[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+1[w−ˉa1)z−ˉc1w+ˉa1ˉc1−ˉb1]α+1+⋯+[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+1×[(w−ˉa1)z−ˉc1w+ˉa1ˉc1−ˉb1]α+2[(w−ˉc1)z−ˉa1w+ˉa1ˉc1−ˉb1]α+2[(w−ˉa2)z−ˉc2w+ˉa2ˉc2−ˉb2]α+2[(w−ˉc2)z−ˉa2w+ˉa2ˉc2−ˉb2]α+2 | (3.7) |
for all w,z∈Π and w≠z. So, from the arbitrariness of w and z in (3.7), we deduce a contradiction. Then, we obtain a2=c2. Similarly, we have a1=c1.
Conversely, if a1=c1 and a2=c2, then it is easy to see that (3.4) holds. This shows that Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J.
We have the following result, and we do not provide proof anymore.
Lemma 3.3. Let τ(z)=1(z−c)α+2 and ϕ(z)=−a−bz−c be the symbols in (I). Then, the operator Wτ,ϕ is complex symmetric on A2α(Π) with the conjugation J if and only if a=c.
Remark 3.1. From Lemma 3.3 and Theorem 3.1, we see that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if both Wτ1,ϕ1 and Wτ2,ϕ2 are complex symmetric on A2α(Π) with the conjugation J.
Example 3.1. Let τ1(z)=1(z−i)α+2, ϕ1(z)=−i+1z−i, τ2(z)=1(z+1−i)α+2, and ϕ2(z)=1−i−1+i2z+2−2i be the symbols in (Ⅰ). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J.
Proof. From the form of the symbols in (Ⅰ), it follows that a1=i, b1=−1, c1=i, a2=−1+i, b2=12+i2 and c2=−1+i. Then, from Theorem 3.1, the desired result follows.
Now, we characterize the complex symmetry of the operator Wτ1,ϕ1−Wτ2,ϕ2 on A2α(Π) with the conjugation Jsf(z)=¯f(ˉz+is).
Theorem 3.2. Let τj(z)=1(z−cj)α+2 and ϕj(z)=−aj−bjz−cj be the symbols in (I) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if a1=c1−is and a2=c2−is.
Proof. For all w,z∈Π, from Lemma 3.2 (a) the following equalities hold
Js(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=Js(1(z−c1)α+22α(α+1)(−a1−b1z−c1+¯w)α+2−1(z−c2)α+22α(α+1)(−a2−b2z−c2+¯w)α+2)=Js(2α(α+1)[(¯w−a1)z+a1c1−b1−c1¯w]α+2−2α(α+1)[(¯w−a2)z+a2c2−b2−c2¯w]α+2)=2α(α+1)[(w−ˉa1)(z−is)+ˉa1ˉc1−ˉb1−ˉc1w]α+2−2α(α+1)[(w−ˉa2)(z−is)+ˉa2ˉc2−ˉb2−ˉc2w]α+2=2α(α+1)[(w−ˉa1)z−(is+ˉc1)w+iˉa1s+ˉa1ˉc1−ˉb1]α+2−2α(α+1)[(w−ˉa2)z−(is+ˉc2)w+iˉa2s+ˉa2ˉc2−ˉb2]α+2 | (3.8) |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗JsKαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(z−is+w)α+2)=(W1(z−ˉa1)α+2,ˉa1ˉc1−ˉb1−ˉc1zz−ˉa1−W1(z−ˉa2)α+2,ˉa2ˉc2−ˉb2−ˉc2zz−ˉa2)(2α(α+1)(z−is+w)α+2)=2α(α+1)[(w−ˉc1−is)z−ˉa1w+ˉa1ˉc1−ˉb1+isˉa1]α+2−2α(α+1)[(w−ˉc2−is)z−ˉa2w+ˉa2ˉc2−ˉb2+isˉa2]α+2. | (3.9) |
Hence, from (3.8), (3.9), and Lemma 3.1, we have that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if
1[(w−ˉa1)z−(is+ˉc1)w+iˉa1s+ˉa1ˉc1−ˉb1]α+2−1[(w−ˉa2)z−(is+ˉc2)w+iˉa2s+ˉa2ˉc2−ˉb2]α+2=1[(w−ˉc1−is)z−ˉa1w+ˉa1ˉc1−ˉb1+isˉa1]α+2−1[(w−ˉc2−is)z−ˉa2w+ˉa2ˉc2−ˉb2+isˉa2]α+2 | (3.10) |
for all w,z∈Π.
Assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js. Then, from (3.10) we obtain
1[(w−ˉa1)z−(is+ˉc1)w+iˉa1s+ˉa1ˉc1−ˉb1]α+2−1[(w−ˉc1−is)z−ˉa1w+ˉa1ˉc1−ˉb1+isˉa1]α+2=1[(w−ˉa2)z−(is+ˉc2)w+iˉa2s+ˉa2ˉc2−ˉb2]α+2−1[(w−ˉc2−is)z−ˉa2w+ˉa2ˉc2−ˉb2+isˉa2]α+2 | (3.11) |
for all w,z∈Π. Also, applying the following formula in (3.11)
xn−yn=(x−y)(xn−1+xn−2y+⋯+xyn−2+yn−1), |
if a2≠c2−is, then
[(w−ˉa2)z−(is+ˉc2)w+iˉa2s+ˉa2ˉc2−ˉb2]α+1+⋯+[(w−ˉc2−is)z−ˉa2w+ˉa2ˉc2−ˉb2+isˉa2]α+1[(w−ˉa1)z−(is+ˉc1)w+iˉa1s+ˉa1ˉc1−ˉb1]α+1+⋯+[(w−ˉc1−is)z−ˉa1w+ˉa1ˉc1−ˉb1+isˉa1]α+1×[(w−ˉa1)z−(is+ˉc1)w+iˉa1s+ˉa1ˉc1−ˉb1]α+2[(w−ˉc1−is)z−ˉa1w+ˉa1ˉc1−ˉb1+isˉa1]α+2[(w−ˉa2)z−(is+ˉc2)w+iˉa2s+ˉa2ˉc2−ˉb2]α+2[(w−ˉc2−is)z−ˉa2w+ˉa2ˉc2−ˉb2+isˉa2]α+2≡ˉc1−ˉa1+isˉc2−ˉa2+is | (3.12) |
for all w,z∈Π with w≠z. So, from the arbitrariness of w and z in (3.12), we deduce a contradiction. Then, we obtain a2=c2−is. Similarly, we have a1=c1−is.
Conversely, if a1=c1−is and a2=c2−is, then we see that (3.10) holds, which shows that Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js.
Remark 3.2. If τ(z)=1(z−c)α+2 and ϕ(z)=−a−bz−c are the symbols in (Ⅰ), then the operator Wτ,ϕ is complex symmetric on A2α(Π) with the conjugation Js if and only if a=c−is. Combining Theorem 3.2, we prove that if τj(z)=1(z−cj)α+2 and ϕj(z)=−aj−bjz−cj are the symbols in (Ⅰ) for j=1, 2, then the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if both Wτ1,ϕ1 and Wτ2,ϕ2 are complex symmetric on A2α(Π) with the conjugation Js.
Example 3.2. Let τ1(z)=1[z−(1+s)i]α+2, ϕ1(z)=−i+1z−(1+s)i, τ2(z)=1[z+1−(1+s)i]α+2, and ϕ2(z)=1−i−1+i2z+2−2(1+s)i be the symbols in (Ⅰ). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js.
Proof. From the form of the symbols in (Ⅰ), it follows that a1=i, b1=−1, c1=(1+s)i, a2=−1+i, b2=12+i2, and c2=−1+(1+s)i. Then, from Theorem 3.2, the desired result follows.
Now, we characterize the complex symmetric operator Wτ1,ϕ1−Wτ2,ϕ2 induced by the symbols in (Ⅰ) on A2α(Π) with the conjugation J∗f(z)=1zα+2¯f(1ˉz).
Theorem 3.3. Let τj(z)=1(z−cj)α+2 and ϕj(z)=−aj−bjz−cj be the symbols in (I) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if a1c1−b1=1 and a2c2−b2=1.
Proof. For all w,z∈Π, from Lemma 3.2 (a) it follows that
J∗(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J∗(1(z−c1)α+22α(α+1)(−a1−b1z−c1+¯w)α+2−1(z−c2)α+22α(α+1)(−a2−b2z−c2+¯w)α+2)=J∗(2α(α+1)[(¯w−a1)z+a1c1−b1−c1¯w]α+2−2α(α+1)[(¯w−a2)z+a2c2−b2−c2¯w]α+2)=2α(α+1)[(ˉa1ˉc1−ˉb1)z−ˉc1wz+w−ˉa1]α+2−2α(α+1)[(ˉa2ˉc2−ˉb2)z−ˉc2wz+w−ˉa2]α+2 | (3.13) |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗J∗Kαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(1+zw)α+2)=(W1(z−ˉa1)α+2,ˉa1ˉc1−ˉb1−ˉc1zz−ˉa1−W1(z−ˉa2)α+2,ˉa2ˉc2−ˉb2−ˉc2zz−ˉa2)(2α(α+1)(1+zw)α+2)=2α(α+1)[z−ˉc1wz+(ˉa1ˉc1−ˉb1)w−ˉa1]α+2−2α(α+1)[z−ˉc2wz+(ˉa2ˉc2−ˉb2)w−ˉa2]α+2. | (3.14) |
Then, from (3.13), (3.14), and Lemma 3.1, it follows that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if
1[(ˉa1ˉc1−ˉb1)z−ˉc1wz+w−ˉa1]α+2−1[(ˉa2ˉc2−ˉb2)z−ˉc2wz+w−ˉa2]α+2=1[z−ˉc1wz+(ˉa1ˉc1−ˉb1)w−ˉa1]α+2−1[z−ˉc2wz+(ˉa2ˉc2−ˉb2)w−ˉa2]α+2 | (3.15) |
for all w,z∈Π. Clearly, (3.15) is equivalent to
1[(ˉa1ˉc1−ˉb1)z−ˉc1wz+w−ˉa1]α+2−1[z−ˉc1wz+(ˉa1ˉc1−ˉb1)w−ˉa1]α+2=1[(ˉa2ˉc2−ˉb2)z−ˉc2wz+w−ˉa2]α+2−1[z−ˉc2wz+(ˉa2ˉc2−ˉb2)w−ˉa2]α+2 |
for all w,z∈Π.
Now, assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. Using the same method in the proof of Theorem 3.1, if a2c2−b2≠1, then
ˉa1ˉc1−ˉb1−1ˉa2ˉc2−ˉb2−1≡[(ˉa2ˉc2−ˉb2)z−ˉc2wz+w−ˉa2]α+1+⋯+[z−ˉc2wz+(ˉa2ˉc2−ˉb2)w−ˉa2]α+1[(ˉa1ˉc1−ˉb1)z−ˉc1wz+w−ˉa1]α+1+⋯+[z−ˉc1wz+(ˉa1ˉc1−ˉb1)w−ˉa1]α+1×[(ˉa1ˉc1−ˉb1)z−ˉc1wz+w−ˉa1]α+2[z−ˉc1wz+(ˉa1ˉc1−ˉb1)w−ˉa1]α+2[(ˉa2ˉc2−ˉb2)z−ˉc2wz+w−ˉa2]α+2[z−ˉc2wz+(ˉa2ˉc2−ˉb2)w−ˉa2]α+2 | (3.16) |
for all w,z∈Π and w≠z. Then, from the arbitrariness of w and z in (3.16), we deduce a contradiction. So, we obtain that a2c2−b2=1. Similarly, we also have that a1c1−b1=1.
Conversely, assume that a1c1−b1=1 and a2c2−b2=1. It is clear that (3.15) holds. This shows that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗.
Remark 3.3. If τ(z)=1(z−c)α+2 and ϕ(z)=−a−bz−c are the symbols in (Ⅰ), then the operator Wτ,ϕ is complex symmetric on A2α(Π) with the conjugation J∗ if and only if ac−b=1. Therefore, from Theorem 3.3, if τj(z)=1(z−cj)α+2 and ϕj(z)=−aj−bjz−cj are the symbols in (Ⅰ) for j=1, 2, then the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if both Wτ1,ϕ1 and Wτ2,ϕ2 are complex symmetric on A2α(Π) with the conjugation J∗.
Example 3.3. Let τ1(z)=1(z−i)α+2, ϕ1(z)=−i+2z−i, τ2(z)=1(z+1+i)α+2 and ϕ2(z)=1−i−1z+1+i be the symbols in (Ⅰ). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗.
Proof. From the form of the symbols in (Ⅰ), it follows that a1=i, b1=−2, c1=i, a2=−1+i, b2=1, and c2=−1−i. From the calculations, we have a1c1−b1=1 and a2c2−b2=1. Then, from Theorem 3.3, the desired result follows.
Next, we characterize the complex symmetric operator Wτ1,ϕ1−Wτ2,ϕ2 induced by the symbols in (Ⅱ) on A2α(Π). First, Lemma 3.2 (b) tells us that
W∗τ,ϕ=Wˉδ(z+ˉμ)α+2,ˉμ−iη=ˉδW1(z+ˉμ)α+2,ˉμ−iη, |
which shows that if a=−μ, b=0 and c=−μ−iη, then
W∗τ,ϕ=ˉδW1(z−ˉa)α+2,ˉaˉc−ˉb−ˉczz−ˉa. |
Therefore, we can directly obtain the following several results.
Theorem 3.4. Let τj(z)=δj(z+μj+iηj)α+2 and ϕj(z)=μj be the symbols in (II) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if η1=η2=0.
Theorem 3.5. Let τj(z)=δj(z+μj+iηj)α+2 and ϕj(z)=μj be the symbols in (II) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if η1=η2=−s.
For the conjugation J∗, we assume that δ≠0. Otherwise, it is trivial.
Theorem 3.6. Let τj(z)=δj(z+μj+iηj)α+2 and ϕj(z)=μj be the symbols in (II) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if
μj=√1−η2j4−iηj2 |
with ηj∈(−2,2), j=1, 2.
Remark 3.4. For the symbols in (Ⅱ), although we do not give the proofs, we still see that the operators Wτ1,ϕ1−Wτ2,ϕ2 are complex symmetric on A2α(Π) with the conjugations J, Js, and J∗ if and only if both Wτ1,ϕ1 and Wτ2,ϕ2 are complex symmetric on A2α(Π) with the conjugations J, Js, and J∗.
Example 3.4. Let τ1(z)=1(z+√32+i2)α+2, ϕ1(z)=√32−i2, τ2(z)=1(z+√154+i4)α+2 and ϕ2(z)=√154−i4 be the symbols in (Ⅱ). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗.
Proof. From the form of the symbols in (Ⅱ), it follows that μ1=√32−i2, η1=1, μ2=√154−i4, and η2=12. Then, from Theorem 3.6, the desired result follows.
Now, we discuss complex the symmetric operator Wτ1,ϕ1−Wτ2,ϕ2 induced by the symbols in (Ⅲ) on A2α(Π).
Theorem 3.7. Let τj(z)=λj and ϕj(z)=z+γj be the symbols in (III) for j=1, 2. Then, Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J.
Proof. For all w,z∈Π, from Lemma 3.2 (c) we have
J(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J(λ12α(α+1)(z+γ1+¯w)α+2−λ22α(α+1)(z+γ2+¯w)α+2)=ˉλ12α(α+1)(z+ˉγ1+w)α+2−ˉλ22α(α+1)(z+ˉγ2+w)α+2 | (3.17) |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗JKαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(z+w)α+2)=(ˉλ1Cz+ˉγ1−ˉλ2Cz+ˉγ2)(2α(α+1)(z+w)α+2)=ˉλ12α(α+1)(z+ˉγ1+w)α+2−ˉλ22α(α+1)(z+ˉγ2+w)α+2. | (3.18) |
From (3.17) and (3.18), it follows that
J(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗JKαw(z) |
for all z∈Π. The proof is completed.
Of course, the following result is true. The proof is omitted.
Theorem 3.8. Let τj(z)=λj and ϕj(z)=z+γj be the symbols in (III) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js.
However, the result on the conjugation J∗ is trivial since there exists the following theorem. Here, assume that λj≠0 for j=1, 2. Otherwise, Wτ1,ϕ1−Wτ2,ϕ2 is a null operator.
Theorem 3.9. Let τj(z)=λj and ϕj(z)=z+γj be the symbols in (III) for j=1, 2. Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if γ1=γ2=0.
Proof. For all w,z∈Π, from Lemma 3.2 (c) it follows that
J∗(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J∗(λ12α(α+1)(z+γ1+¯w)α+2−λ22α(α+1)(z+γ2+¯w)α+2)=ˉλ12α(α+1)(1+wz+ˉγ1z)α+2−ˉλ22α(α+1)(1+wz+ˉγ2z)α+2 | (3.19) |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗J∗Kαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(1+zw)α+2)=(ˉλ1Cz+ˉγ1−ˉλ2Cz+ˉγ2)(2α(α+1)(1+zw)α+2)=ˉλ12α(α+1)(1+wz+ˉγ1w)α+2−ˉλ22α(α+1)(1+wz+ˉγ2w)α+2. | (3.20) |
Then, from (3.19), (3.20), and Lemma 3.1, it follows that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if
ˉλ1[1(1+wz+ˉγ1z)α+2−1(1+wz+ˉγ1w)α+2]=ˉλ2[1(1+wz+ˉγ2z)α+2−1(1+wz+ˉγ2w)α+2] | (3.21) |
for all w,z∈Π.
Now, assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. Using the above-mentioned method in the proof of Theorem 3.1, if γ2≠0, then
ˉλ1ˉγ1ˉλ2ˉγ2≡(1+wz+ˉγ2w)α+1+⋯+(1+wz+ˉγ2z)α+1(1+wz+ˉγ1w)α+1+⋯+(1+wz+ˉγ1z)α+1(1+wz+ˉγ1z)α+2(1+wz+ˉγ1w)α+2(1+wz+ˉγ2z)α+2(1+wz+ˉγ2w)α+2 | (3.22) |
for all w,z∈Π and w≠z. Then, from the arbitrariness of w and z in (3.22), we deduce a contradiction. So, we obtain that γ2=0. Similarly, we also have that γ1=0.
Conversely, assume that γ1=γ2=0. It is clear that (3.21) holds. This shows that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗.
In this section, we first consider complex symmetric difference induced by the symbols in (Ⅰ) and (Ⅱ). Assume that δ≠0. Otherwise, Wτ1,ϕ1−Wτ2,ϕ2=Wτ1,ϕ1 in Theorem 4.1, whose complex symmetry has been studied in Lemma 3.3.
Theorem 4.1. Let τ1(z)=1(z−c)α+2 and ϕ1(z)=−a−bz−c be the symbols in (I), τ2(z)=δ(z+μ+iη)α+2 and ϕ2(z)=μ the symbols in (II). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if a=c and η=0.
Proof. For all w,z∈Π, it follows from Lemma 3.2 (a) and (b) that
J(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J(1(z−c)α+22α(α+1)(−a−bz−c+¯w)α+2−δ(z+μ+iη)α+22α(α+1)(¯w+μ)α+2)=J(2α(α+1)[(¯w−a)z+ac−b−c¯w]α+2−2α(α+1)δ[(¯w+μ)z+(μ+iη)¯w+μ2+(iη)μ]α+2)=2α(α+1)[(w−ˉa)z+ˉaˉc−ˉb−ˉcw]α+2−2α(α+1)ˉδ[(w+ˉμ)z+(ˉμ−iη)w+ˉμ2−(iη)ˉμ]α+2 | (4.1) |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗JKαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(z+w)α+2)=(W1(z−ˉa)α+2,ˉaˉc−ˉb−ˉczz−ˉa−ˉδW1(z+ˉμ)α+2,ˉμ−iη)(2α(α+1)(z+w)α+2)=2α(α+1)[(w−ˉc)z−ˉaw+ˉaˉc−ˉb]α+2−2α(α+1)ˉδ[(ˉμ−iη+w)z+ˉμw+ˉμ2−(iη)ˉμ]α+2. | (4.2) |
Therefore, by Lemma 3.1, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if
1[(w−ˉa)z+ˉaˉc−ˉb−ˉcw]α+2−1[(w−ˉc)z−ˉaw+ˉaˉc−ˉb]α+2=ˉδ[(w+ˉμ)z+(ˉμ−iη)w+ˉμ2−iηˉμ]α+2−ˉδ[(ˉμ−iη+w)z+ˉμw+ˉμ2−iηˉμ]α+2 | (4.3) |
for all w,z∈Π.
Assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J. Then, by the formula
xn−yn=(x−y)(xn−1+xn−2y+⋯+xyn−2+yn−1), |
(4.3) becomes
(ˉa−ˉc)(z−w){[(w−ˉa)z+ˉaˉc−ˉb−ˉcw]α+1+⋯+[(w−ˉc)z−ˉaw+ˉaˉc−ˉb]α+1}[(w−ˉa)z+ˉaˉc−ˉb−ˉcw]α+2[(w−ˉc)z−ˉaw+ˉaˉc−ˉb]α+2=iηˉδ(w−z){[(w+ˉμ)z+(ˉμ−iη)w+ˉμ2−iηˉμ]α+1+⋯+[(ˉμ−iη+w)z+ˉμw+ˉμ2−iηˉμ]α+1}[(w+ˉμ)z+(ˉμ−iη)w+ˉμ2−iηˉμ]α+2[(ˉμ−iη+w)z+ˉμw+ˉμ2−iηˉμ]α+2 | (4.4) |
for all w,z∈Π. Similar to the proof of Theorem 3.1, we see that if a≠c, then we can deduce a contradiction. Similarly, we can obtain η=0.
Conversely, if a=c and η=0, then it is easy to see that (4.3) holds. By Lemma 3.1, Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J. The proof is completed.
Example 4.1. Let τ1(z)=1(z+2−i)α+2 and ϕ1(z)=2−i−1z+2−i be the symbols in (Ⅰ), τ2(z)=1(z+√32−i2)α+2 and ϕ2(z)=√32−i2 be the symbols in (Ⅱ). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J.
Proof. It is clear that a=c=−2+i, μ=√32−i2 and η=0. From Theorem 4.1, the desired result follows.
Theorem 4.2. Let τ1(z)=1(z−c)α+2 and ϕ1(z)=−a−bz−c be the symbols in (I), τ2(z)=λ and ϕ2(z)=z+γ the symbols in (III). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if a=c.
Proof. For all w,z∈Π, it follows from Lemma 3.2 (a) and (c) that
J(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J(1(z−c)α+22α(α+1)(−a−bz−c+¯w)α+2−λ2α(α+1)(z+γ+¯w)α+2)=J(2α(α+1)[(¯w−a)z+ac−b−c¯w]α+2−λ2α(α+1)(z+γ+¯w)α+2)=2α(α+1)[(w−ˉa)z+ˉaˉc−ˉb−ˉcw]α+2−ˉλ2α(α+1)(z+ˉγ+w)α+2 |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗JKαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(z+w)α+2)=(W1(z−ˉa)α+2,ˉaˉc−ˉb−ˉczz−ˉa−ˉλCz+ˉγ)(2α(α+1)(z+w)α+2)=2α(α+1)[(w−ˉc)z−ˉaw+ˉaˉc−ˉb]α+2−ˉλ2α(α+1)(z+ˉγ+w)α+2. |
Therefore, by Lemma 3.1, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if
1[(w−ˉa)z+ˉaˉc−ˉb−ˉcw]α+2=1[(w−ˉc)z−ˉaw+ˉaˉc−ˉb]α+2 | (4.5) |
for all w,z∈Π.
Assume that Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J. Then, from (4.4) we obtain
(w−ˉa)z+ˉaˉc−ˉb−ˉcw=(w−ˉc)z−ˉaw+ˉaˉc−ˉb |
for all w,z∈Π, that is, a=c.
Conversely, if a=c, then it is clear that (4.5) holds. By Lemma 3.1, Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J. The proof is completed.
Theorem 4.3. Let τ1(z)=δ(z+μ+iη)α+2 and ϕ1(z)=μ be the symbols in (II), τ2(z)=λ and ϕ2(z)=z+γ the symbols in (III). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if η=0.
Proof. For w,z∈Π, by Lemma 3.2 (b) and (c),
J(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J(δ(z+μ+iη)α+22α(α+1)(¯w+μ)α+2−λ2α(α+1)(z+γ+¯w)α+2)=2α(α+1)ˉδ[(w+ˉμ)z+(ˉμ−iη)w+ˉμ2−(iη)ˉμ]α+2−ˉλ2α(α+1)(z+ˉγ+w)α+2 |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗JKαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(z+w)α+2)=(ˉδW1(z+ˉμ)α+2,ˉμ−iη−ˉλCz+ˉγ)(2α(α+1)(z+w)α+2)=2α(α+1)ˉδ[(ˉμ−iη+w)z+ˉμw+ˉμ2−(iη)ˉμ]α+2−ˉλ2α(α+1)(z+ˉγ+w)α+2. |
Thus, from Lemma 3.1, it follows that Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J if and only if η=0.
Next, we do not give the examples since one can easily give examples.
Theorem 4.4. Let τ1(z)=1(z−c)α+2 and ϕ1(z)=−a−bz−c be the symbols in (I), τ2(z)=δ(z+μ+iη)α+2 and ϕ2(z)=μ the symbols in (II). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if ac−b=1 and μ=√1−η24−η2i, where η∈(−2,2).
Proof. For all w,z∈Π, from Lemma 3.2 (a) and (b), it follows that
J∗(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J∗(2α(α+1)[(¯w−a)z+ac−b−c¯w]α+2−2α(α+1)δ[(¯w+μ)z+(μ+iη)¯w+μ2+iημ]α+2)=2α(α+1)[(ˉaˉc−ˉb)z−ˉcwz+w−ˉa]α+2−2α(α+1)ˉδ[(ˉμ−iη)wz+(ˉμ2−iηˉμ)z+w+ˉμ]α+2 |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗J∗Kαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(1+zw)α+2)=(W1(z−ˉa)α+2,ˉaˉc−ˉb−ˉczz−ˉa−ˉδW1(z+ˉμ)α+2,ˉμ−iη)(2α(α+1)(1+zw)α+2)=2α(α+1)[z−ˉcwz+(ˉaˉc−ˉb)w−ˉa]α+2−2α(α+1)ˉδ[(ˉμ−iη)wz+z+(ˉμ2−iηˉμ)w+ˉμ]α+2. |
By Lemma 3.1, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if
1[(ˉaˉc−ˉb)z−ˉcwz+w−ˉa]α+2−1[z−ˉcwz+(ˉaˉc−ˉb)w−ˉa]α+2=ˉδ[(ˉμ−iη)wz+(ˉμ2−iηˉμ)z+w+ˉμ]α+2−ˉδ[(ˉμ−iη)wz+z+(ˉμ2−iηˉμ)w+ˉμ]α+2 | (4.6) |
for all w,z∈Π.
Assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. By using the same method, we obtain that ac−b=1 and μ=√1−η24−η2i, where η∈(−2,2).
Conversely, if ac−b=1 and μ=√1−η24−η2i, where η∈(−2,2), then it is clear that (4.6) holds, which shows the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. The proof is completed.
Theorem 4.5. Let τ1(z)=1(z−c)α+2 and ϕ1(z)=−a−bz−c be the symbols in (I), τ2(z)=λ and ϕ2(z)=z+γ the symbols in (III). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if ac−b=1 and γ=0.
Proof. For all w,z∈Π, from Lemma 3.2 (a) and (c), it follows that
J∗(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J∗(2α(α+1)[(¯w−a)z+ac−b−c¯w]α+2−2α(α+1)λ(¯w+z+γ)α+2)=2α(α+1)[(ˉaˉc−ˉb)z−ˉcwz+w−ˉa]α+2−2α(α+1)ˉλ(wz+ˉγz+1)α+2 |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗J∗Kαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(1+zw)α+2)=(W1(z−ˉa)α+2ˉaˉc−ˉb−ˉczz−ˉa−ˉλCz+ˉγ)(2α(α+1)(1+zw)α+2)=2α(α+1)[z−ˉcwz+(ˉaˉc−ˉb)w−ˉa]α+2−2α(α+1)ˉλ(wz+ˉγw+1)α+2. |
By Lemma 3.1, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if
1[(ˉaˉc−ˉb)z−ˉcwz+w−ˉa]α+2−1[z−ˉcwz+(ˉaˉc−ˉb)w−ˉa]α+2 =ˉλ(wz+ˉγz+1)α+2−ˉλ(wz+ˉγw+1)α+2 | (4.7) |
for all w,z∈Π.
Assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. By using the same method, we obtain that ac−b=1 and γ=0.
Conversely, if ac−b=1 and γ=0, then it is clear that (4.8) holds, which shows the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. The proof is completed.
Theorem 4.6. Let τ1(z)=δ(z+μ+iη)α+2 and ϕ1(z)=μ the symbols in (II), τ2(z)=λ and ϕ2(z)=z+γ the symbols in (III). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if γ=0 and μ=√1−η24−η2i, where η∈(−2,2).
Proof. For all w,z∈Π, from Lemma 3.2 (b) and (c), it follows that
J∗(Wτ1,ϕ1−Wτ2,ϕ2)Kαw(z)=J∗(2α(α+1)δ[(¯w+μ)z+(μ+iη)¯w+μ2+iημ]α+2−2α(α+1)λ(¯w+z+γ)α+2)=2α(α+1)ˉδ[(ˉμ−iη)wz+(ˉμ2−iηˉμ)z+w+ˉμ]α+2−2α(α+1)ˉλ(wz+ˉγz+1)α+2 |
and
(Wτ1,ϕ1−Wτ2,ϕ2)∗J∗Kαw(z)=(Wτ1,ϕ1−Wτ2,ϕ2)∗(2α(α+1)(1+zw)α+2)=(ˉδW1(z+ˉμ)α+2,ˉμ−iη−ˉλCz+ˉγ)(2α(α+1)(1+zw)α+2)=2α(α+1)ˉδ[(ˉμ−iη)wz+z+(ˉμ2−iηˉμ)w+ˉμ]α+2−2α(α+1)ˉλ(wz+ˉγw+1)α+2. |
By Lemma 3.1, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗ if and only if
1[(ˉaˉc−ˉb)z−ˉcwz+w−ˉa]α+2−1[z−ˉcwz+(ˉaˉc−ˉb)w−ˉa]α+2=ˉλ(wz+ˉγz+1)α+2−ˉλ(wz+ˉγw+1)α+2 | (4.8) |
for all w,z∈Π.
Assume that the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. By using the same method, we obtain that ac−b=1 and γ=0.
Conversely, if ac−b=1 and γ=0, then it is clear that (4.8) holds, which shows the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J∗. The proof is completed.
From the above proofs, the following results can be similarly proved.
Theorem 4.7. Let τ1(z)=1(z−c)α+2 and ϕ1(z)=−a−bz−c be the symbols in (I), τ2(z)=δ(z+μ+iη)α+2 and ϕ2(z)=μ the symbols in (II). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if a=c−is and η=−s.
Theorem 4.8. Let τ1(z)=1(z−c)α+2 and ϕ1(z)=−a−bz−c be the symbols in (I), τ2(z)=λ and ϕ2(z)=z+γ the symbols in (III). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if a=c−is.
Theorem 4.9. Let τ1(z)=δ(z+μ+iη)α+2 and ϕ1(z)=μ be the symbols in (II), τ2(z)=λ and ϕ2(z)=z+γ the symbols in (III). Then, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation Js if and only if η=−s.
Remark 4.1. Considering the results in Section 3, the operator Wτ1,ϕ1−Wτ2,ϕ2 is complex symmetric on A2α(Π) with the conjugation J, Js, and J∗, respectively, if and only if both Wτ1,ϕ1 and Wτ2,ϕ2 are complex symmetric on A2α(Π) with the conjugation J, Js, and J∗, respectively.
Since it is impossible to give the proper description of the adjoint of the operator Wτ,ϕ with the general symbols on A2α(Π), in this paper we just consider this problem for the operators Wτ,ϕ with the symbols in (Ⅰ)–(Ⅲ) on A2α(Π). At the same time, by using these descriptions, we characterize complex symmetric difference of the operators Wτ,ϕ with the symbols in (Ⅰ)–(Ⅲ) with the conjugations J, Js, and J∗ on A2α(Π). However, we still do not obtain any result for the general symbols. Therefore, we hope that the study can attract more attention for such a topic.
The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.
The author thanks the anonymous referees for their time and comments.
This study was supported by Sichuan Science and Technology Program (2024NSFSC2314) and the Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering (SUSE652B002).
The author declares that he has no competing interests.
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1. | Zhi-Jie Jiang, 3-Complex Symmetric and Complex Normal Weighted Composition Operators on the Weighted Bergman Spaces of the Half-Plane, 2024, 12, 2227-7390, 980, 10.3390/math12070980 |