Interplay between complex symmetry and Koenigs eigenfunctions
aa r X i v : . [ m a t h . F A ] S e p Interplay between complex symmetry and Koenigs eigenfunctions
S. Waleed Noor ∗ , Osmar R. Severiano IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil.
Abstract
We investigate the relationship between the complex symmetry of composition operators C φ f = f ◦ φ induced on the classical Hardy space H ( D ) by an analytic self-map φ of theopen unit disk D and its Koenigs eigenfunction. A generalization of orthogonality known as conjugate-orthogonality will play a key role in this work. We show that if φ is a Schr¨odermap (fixes a point a ∈ D with 0 < | φ ′ ( a ) | <
1) and σ is its Koenigs eigenfunction, then C φ is complex symmetric if and only if ( σ n ) n ∈ N is complete and conjugate-orthogonal in H ( D ).We study the conjugate-orthogonality of Koenigs sequences with some concrete examples.We use these results to show that commutants of complex symmetric composition operatorswith Schr¨oder symbols consist entirely of complex symmetric operators. Keywords:
Complex symmetric operator, composition operator, Koenigs eigenfunction.
1. Introduction
Let X be a vector space of analytic functions on the open unit disk D and φ an analyticself-map of D . The composition operator C φ with symbol φ is defined as C φ f = f ◦ φ for f ∈ X. Operators of this type have been studied on a variety of spaces and in great detail. Theobjective is to study the interaction between the operator theortic properties of C φ andfunction theoretic properties of φ . It is well-known that C φ is bounded on the classical HardyHilbert space H ( D ) whenever φ is an analytic self-map of D . The monographs of Shapiro[21] and Cowen and McCluer [5] contain detailed accounts of the subject. An analytic selfmap φ of D is called a Schr¨oder map if it fixes a point a ∈ D with 0 < | φ ′ ( a ) | <
1. Let H ( D ) denote Frechet space of all analytic functions on D . In 1884, Koenigs [13] showedthat if φ is a Schr¨oder map with fixed point a ∈ D , then the eigenvalues of the operator C φ : H ( D ) → H ( D ) are 1 , φ ′ ( a ) , φ ′ ( a ) , φ ′ ( a ) , . . . . ∗ Corresponding author. Telephone: +55 19 992360945.
Email addresses: [email protected] (S. Waleed Noor), [email protected] (Osmar R. Severiano)
Preprint submitted to Elsevier September 17, 2020 hese eigenvalues are simple (that have multiplicity one) and the eigenfunction σ cor-responding to the eigenvalue φ ′ ( a ) is called the Koenigs eigenfunction for φ . It follows that σ n is an eigenfunction corresponding to φ ′ ( a ) n for each n ∈ N , and all such eigenfunctionsare scalar multiples of σ n . The sequence ( σ n ) n ∈ N will be referred to as the Koenigs sequence for φ . The eigenfunction σ does not necessarily belong to any of the Hardy spaces H p ( D )for p >
0. Characterizations of H p ( D )-membership of σ have been obtained by Bourdonand Shapiro [2, 3] and Poggi-Corradini [19]. In particular, they show that σ ∈ H p ( D ) for all p >
0, or equivalently σ n ∈ H ( D ) for all n ∈ N if and only if the composition operator C φ is a Riesz operator , that is when the essential spectrum of C φ is the singleton { } .On the other hand a bounded operator T on a separable Hilbert space H is complexsymmetric if T has a self-transpose matrix representation with respect to some orthonormalbasis for H . An equivalent definition also exists. A conjugation is a conjugate-linear opera-tor J : H → H that satisfies the conditions(a) J is isometric : h J f, J g i = h g, f i ∀ f, g ∈ H ,(b) J is involutive : J = I .Then T is J - symmetric if J T = T ∗ J , and is called complex symmetric if T is J -symmetricfor some conjugation J on H . A sequence ( f n ) n ∈ N is conjugate-orthogonal in H if thereexists a conjugation J on H such that h J f n , f m i = 0 for all n = m . An orthonormal basis( e n ) n ∈ N for H is always conjugate-orthogonal. Indeed, just define the conjugation J e n = e n for all n ∈ N and extend to all of H by conjugate-linearity. Hence h J e n , e m i = δ m,n where δ m,n is the Kronecker delta. Complex symmetric operators are natural generalizations ofcomplex symmetric matrices and normal operators, and their general study was initiated byGarcia, Putinar, and Wogen ([7],[8],[11],[12]). Conjugate-orthogonal systems were studiedby Garcia and Putinar ([9][10]) as eigenvectors of complex symmetric operators.The study of complex symmetric composition operators on H ( D ) was initiated by Garciaand Hammond [6]. They showed that involutive disk automorphisms induce non-normal complex symmetric composition operators. This was the first such example. Bourdon andNoor [1] proved that if C φ is complex symmetric then φ must necessarily fix a point in D . They further showed that if φ is a disk automorphism that is not elliptic of order two(involutive) or three, then C φ is never complex sysmmetric. Then Narayan, Sievewright andThompson [15] discovered the first non-automorphic composition operators that are complexsymmetric. Recently Narayan, Sievewright, Tjani [16] characterized all non-automorphiclinear factional φ for which C φ is complex symmetric. The analogous problem for the Hardyspace of the half-plane H ( C + ) was recently solved by Noor and Severiano [18]. For moregeneral symbols the problem still remains open. It follows that the non-automorphic symbols φ of interest from the point of view of complex symmetry are those that fix a point in D ,and that are also univalent (see [6, Prop. 2.5]). These φ are precisely the univalent Scr¨odermaps and the Koenigs eigenfunction σ for φ is also univalent. In this case if C φ is complexsymmetric then it is a Riesz operator and the Koenigs sequence ( σ n ) n ∈ N is contained in H ( D ) ([6, Prop. 2.7]). 2he objective of this article is to study properties of the Koenigs sequence ( σ n ) n ∈ N andhow they interact with the complex symmetry of C φ . The main result of this article cannow be stated as follows (see Theorem 3.1). Theorem 1.1.
Let φ be a Schr¨oder map with Koenigs eigenfunction σ . Then C φ is complexsymmetric if and only if the Koenigs sequence ( σ n ) n ∈ N is complete and conjugate-orthogonalin H ( D ) . A Riesz basis is similar to an orthonormal basis (ONB), that is, it is the image of an ONBunder an invertible operator. But Riesz bases are not necessarily conjugate-orthogonal. Let φ a ( z ) = a − z − ¯ az be the involutive disk automorphism for some a ∈ D . Then ( φ na ) n ∈ N is a Rieszbasis and is the Koenigs sequence for the Schr¨oder mapΦ a,λ ( z ) = φ a ( λφ a ( z ))with fixed point a and λ ∈ D . We then apply Theorem 1.1 to show that this Riesz basis( φ na ) n ∈ N is not conjugate-orthogonal (see Theorem 4.1). Theorem 1.2.
For a, λ ∈ D \ { } , the composition operator C Φ a,λ is not complex symmetricon H ( D ) , and therefore ( φ na ) n ∈ N is not conjugate-orthogonal. We next provide an example of a complete and conjugate-orthogonal sequence that is not an ONB in H ( D ) (see Theorem 5.2). Theorem 1.3.
The sequence (( z − a ) n ) n ∈ N is complete and conjugate-orthogonal for all a ∈ D , and is not an ONB when a = 0 . If φ ( z ) = cz + d is an affine self-map of D with fixed point a ∈ D , then (( z − a ) n ) n ∈ N is theKoenigs sequence for φ . Therefore by Theorems 1.1 and 1.3 we get a simple proof for oneof the main results of Narayan, Sievewright and Thompson [15] (see Corollary 5.3). Corollary 1.4. If φ is an affine self-map with a fixed point in D , then C φ is complex sym-metric on H ( D ) . The set of bounded operators that commute with an operator T is called the commutant of T and is denoted by T ′ . Our main result about commutants of complex symmetriccomposition operators is the following (see Theorem 6.1). Theorem 1.5.
Let φ be a Schr¨oder map such that C φ is complex symmetric. Then each A ∈ C ′ φ is also complex symmetric. This is not true in general. If φ is an elliptic automorphism of order 2, then C φ is complexsymmetric (see [6] and [17]), but C ′ φ contains composition operators that are not complexsymmetric. On the other hand if T is complex symmetric, then T n for n ∈ N are alsocomplex symmetric. For composition operators with Schr¨oder symbols we obtain a strongconverse of this (see Corollary 6.2). 3 orollary 1.6. Let φ be a Schr¨oder map. Then C φ is complex symmetric if and only if C nφ is complex symmetric for some integer n ≥ . This is not true in general even for composition operators, since if φ is an elliptic auto-morphism of order 4 then C φ is not complex symmetric (see [1, Prop. 3.3]), but C φ is ellipticof order two and is hence complex symmetric.
2. Preliminaries H ( D )The Hardy space H := H ( D ) is the space of all analytic functions in D given by f ( z ) = P ∞ n =0 b f ( n ) z n , for which the norm k f k := ∞ X n =0 | b f ( n ) | < ∞ . The Hardy space H is a Hilbert space with the following inner product: h f, g i = ∞ X n =0 b f ( n ) b g ( n )where f, g ∈ H . For each non-negative integer n and a ∈ D , the evaluation of the n thderivative of f ∈ H at a is f ( n ) ( a ) = h f, K ( n ) a i where K ( n ) a ( z ) = ∞ X m = n m !( m − n )! a m − n z m . (1)We write f = f (0) and hence K a ( z ) = K (0) a ( z ) = − ¯ az is the usual reproducing kernel for H at a ∈ D . Let φ be a Schr¨oder map with fixed point a ∈ D and σ its Koenigs eigenfunctionnormalized with || σ || = 1. If C φ is complex symmetric, then Garcia and Hammond [6, p.176]proved the following relation we will use | σ (0) | = | K (1) a ( a ) |k K a k k K (1) a k . (2) Complex symmetric operators satisfy the following spectral symmetry property . Let T be J -symmetric on a Hilbert space H and λ ∈ C , then f ∈ Ker( T − λI ) ⇐⇒ J f ∈ Ker( T ∗ − λI ) . (3)A sequence ( f n ) n ∈ N is conjugate-orthogonal in H if h J f n , f m i = 0 for n = m and someconjugation J on H . Alternatively we shall just say ( f n ) n ∈ N is J -orthogonal. Note that if( f n ) n ∈ N is also complete in H , then h J f n , f n i 6 = 0 for all n ∈ N . This implies that ( J f n ) n ∈ N isthe unique sequence (upto scalar multiples) that is biorthogonal to ( f n ) n ∈ N (see [4, Lemma3.3.1]. In [10], Garcia proved the following. 4 emma 2.1. The eigenvectors of a J -symmetric operator T corresponding to distinct eigen-values are J -orthogonal. Proof.
Let
T f i = λ i f i for i = 1 , λ = λ . Then λ h J f , f i = h J T f , f i = h T ∗ J f , f i = h J f , T f i = λ h J f , f i implies that h J f , f i = 0.If φ is a Schr¨oder map with fixed point a ∈ D and λ = φ ′ ( a ), then the eigenvalues ( λ n ) n ∈ N of C φ are all distinct since | λ | <
1. Therefore Lemma 2.1 gives us
Proposition 2.2.
Let φ be a Schr¨oder map with Koenigs eigenfunction σ. If C φ is complexsymmetric then ( σ n ) n ∈ N is conjugate-orthogonal in H .
3. Complex symmetry and Koenigs eigenfuctions
In this section, we characterize the complex symmetry of C φ with Schr¨oder symbol φ interms of the conjugate-orthogonality of its Koenigs sequence ( σ n ) n ∈ N . Theorem 3.1.
Let φ be a Schr¨oder map with Koenigs eigenfunction σ . Then C φ is complexsymmetric if and only if the Koenigs sequence ( σ n ) n ∈ N is complete and conjugate-orthogonalin H . Proof.
Let a ∈ D be the fixed point of φ and denote λ = φ ′ ( a ). First suppose ( σ n ) n ∈ N iscomplete and J -orthogonal. Then for n, m ∈ N , we have h ( C ∗ φ J − J C φ ) σ n , σ m i = h J σ n , C φ σ m i − ¯ λ n h J σ n , σ m i = (¯ λ m − ¯ λ n ) h J σ n , σ m i . Since h J σ n , σ m i = 0 for all n = m , the completeness of ( σ n ) n ∈ N implies that C ∗ φ J = J C φ , andtherefore that C φ is J -symmetric. Conversely suppose C φ is J -symmetric. By Proposition2.2 we know that ( σ n ) n ∈ N is conjugate-orthogonal. Hence we only need to show that ( σ n ) n ∈ N is complete in H . For each n ∈ N , the function K ( n ) a can be written in the form K ( n ) a = a n v n + a n − v n − + . . . + a v , (4)where v j is an eigenvector for C ∗ φ corresponding to the eigenvalue λ j , and the a j are scalars(see [6, Prop. 2.6]). Now apply J on both sides of (4) and note that each J v j is a scalarmultiple of σ j by spectral symmetry (see (3)) and the fact that the λ j are simple eigenvaluesof C φ . We then get J K ( n ) a = c n σ n + c n − σ n − + . . . + c σ. To conclude the completeness of ( σ n ) n ∈ N , we note that if some f ∈ H is orthogonal to( σ n ) n ∈ N then f ⊥ J K ( n ) a and hence J f ⊥ K ( n ) a for all n ∈ N . But ( J f ) ( n ) ( a ) = 0 for all n implies J f ≡ f ≡
0. Therefore ( σ n ) n ∈ N is complete in H .5 . A Riesz basis that is not conjugate-orthogonal Let φ a ( z ) = a − z − ¯ az be the involutive disk automorphism for some a ∈ D \ { } . The goalof this section is to prove that although ( φ na ) n ∈ N is a Koenigs sequence and a Riesz basis,it is not conjugate-orthogonal. The sequence ( φ na ) n ∈ N is a Riesz basis since φ na = C φ a z n for n ∈ N and it is also the Koenigs sequence for the Schr¨oder mapΦ a,λ ( z ) = φ a ( λφ a ( z ))since C Φ a,λ φ a = λφ a with λ ∈ D \ { } and fixed point a . Theorem 4.1.
For a, λ ∈ D \ { } , the composition operator C Φ a,λ is not complex symmetricon H , and therefore ( φ na ) n ∈ N is not conjugate-orthogonal. Proof.
First let 0 < a <
1. Observe that by (1) we have K (1) a ( z ) = ∞ X n =1 na n − z n . Since φ a is inner we have || φ a || = 1. Moreover, simple computations show that | K (1) a ( a ) | = a (1 − a ) and k K (1) a k = ( a + 1) / (1 − a ) / . (5)Now suppose on the contrary that C Φ a,λ is complex symmetric. Then (2) with σ = φ a and φ = Φ a,λ gives | K (1) a ( a ) | = | φ a (0) | k K a k k K (1) a k . (6)By combining (5) and (6), we obtain a ( a − = a (1 + a ) / ( a − which is possible only if a = 0 . However, this contradicts the hypothesis that a ∈ (0 ,
1) andtherefore C Φ a,λ is not complex symmetric. For the general case a ∈ D \ { } , let θ be a realnumber such that a = | a | e iθ . Then e − iθ (cid:0) φ a (cid:0) e iθ z (cid:1)(cid:1) = e − iθ (cid:18) a − e iθ z − ae iθ z (cid:19) = | a | − z − | a | z = φ | a | ( z ) . (7)By using (7), we can express Φ a,λ in term of Φ | a | ,λ e − iθ Φ a,λ ( e iθ z ) = e − iθ φ α ( λφ α ( e iθ z )) = e − iθ φ α ( λe iθ e − iθ φ α ( e iθ z ))= e − iθ φ α ( e iθ λφ | α | ( z )) = φ | α | ( λφ | α | ( z ))= Φ | a | ,λ ( z ) . This implies that C e iθ z C Φ a,λ C ∗ e iθ z = C Φ | a | ,λ . Since C e iθ z is a unitary operator on H , it followsthat C Φ a,λ is unitarily equivalent to C Φ | a | ,λ and therefore C Φ a,λ is not complex symmetric.Finally since ( φ na ) n ∈ N is the Koenigs sequence for Φ a,λ and is complete in H , by Theorem3.1 it follows that ( φ na ) n ∈ N is not conjugate-orthogonal.6 . A complete and conjugate-orthogonal Koenigs sequence Let a ∈ D and σ ( z ) = z − a . In this section we show that ( σ n ) n ∈ N is a complete andconjugate-orthogonal sequence that is not an ONB when a = 0. First recall that for r ∈ R and j ∈ N , the generalized binomial coefficient is defined as (cid:18) rj (cid:19) = r ( r − . . . ( r − ( j − j !if j > (cid:0) rj (cid:1) = 1 if j = 0. If r ∈ N with 0 ≤ j ≤ r , then (cid:0) rj (cid:1) = r ! j !( r − j )! is the usualbinomial coefficient. In general, the binomial series for (1 − z ) r is(1 − z ) r = ∞ X j =0 (cid:18) rj (cid:19) ( − z ) j (8)for each z ∈ D . We begin by showing that ( σ n ) n ∈ N has a biorthogonal sequence ( z n K n +1 a ) n ∈ N when a ∈ ( − , Lemma 5.1.
For each a ∈ ( − , and non-negative integers n and m, we have h z n K n +1 a , ( z − a ) m i = δ nm , (9) where δ is the Kronecker delta function. Proof.
First observe that K n +1 a ( z ) = (1 − az ) − ( n +1) . By using (8), we get z n K n +1 a ( z ) = ∞ X j =0 (cid:18) − n − j (cid:19) ( − a ) j z j + n . (10)Then (9) holds for m ≤ n since the smallest term in (10) is z n , whereas the largest term in( z − a ) m is z m . Therefore let m > n and defining k = m − n we get h z n K n +1 a , ( z − a ) m i = h ∞ X j =0 (cid:18) − n − j (cid:19) ( − a ) j z j + n , m X j =0 (cid:18) mj (cid:19) ( − a ) m − j z j i = m − n X j =0 (cid:18) − n − j (cid:19)(cid:18) mn + j (cid:19) ( − a ) m − n = ( − a ) k k X j =0 ( − n − − n − · · · ( − n − j ) j ! m !( n + j )!( k − j )!= ( − a ) k k X j =0 ( − j ( n + 1)( n + 2) · · · ( n + j ) j ! m !( n + j )!( k − j )!= ( − a ) k k X j =0 ( − j ( n + j )! j ! n ! m !( n + j )!( k − j )!= ( − a ) k m ! n ! k ! k X j =0 ( − j (cid:18) kj (cid:19) = 0 . P kj =0 ( − j (cid:0) kj (cid:1) = 0 for k > J be the conjugation on H defined by ( J f )( z ) = f (¯ z ), T a the operator of multi-plication by the normalized reproducing kernel K a / k K a k and φ a ( z ) = a − z − ¯ az . If a ∈ ( − , J a = J T a C φ a is a conjugation on H by [15, Prop. 2.3]. We arrive at the main result. Theorem 5.2. If a ∈ D , then (( z − a ) n ) n ∈ N is complete and conjugate-orthogonal in H . Itis not an ONB when a = 0 . Proof.
First suppose a ∈ ( − , φ a ( z ) − a = − (1 − a ) zK a ( z ) = − zK a ( z ) / || K a || . Then we get J a ( z − a ) n = K a ( z )( φ a ( z ) − a ) n / || K a || = ( − n || K a || − n − z n K n +1 a ( z ) . (11)Therefore J a ( z − a ) n is a scalar multiple of z n K n +1 a for n ∈ N and hence ( z − a ) n is J a -orthogonal by Lemma 5.1 for a ∈ ( − , a ∈ D , let θ be a real numbersuch that a = e iθ | a | . Since J | a | is a conjugation for each a ∈ D and U θ = C e iθ z is a unitaryoperator, it follows that U ∗ θ J | a | U θ is also a conjugation. Therefore we get h U ∗ θ J | a | U θ ( z − a ) n , ( z − a ) m i = h J | a | U θ ( z − a ) n , U θ ( z − a ) m i = h J | a | ( e iθ z − a ) n , ( e iθ z − a ) m i = e − iθ ( n + m ) h J | a | ( z − | a | ) n , ( z − | a | ) m i for m, n ∈ N . Now since ( z − | a | ) n is J | a | -orthogonal by the first part, it follows that(( z − a ) n ) n ∈ N is U ∗ θ J | a | U θ -orthogonal. We now prove the completeness of (( z − a ) n ) n ∈ N . Notethat the operator U : f ( z ) → f ( z − a ) is unbounded on H and has the polynomials C [ z ]in its domain. By U z n = ( z − a ) n for n ∈ N it is enough to show that U C [ z ] = C [ z ]. Thisis clear since for any polynomial p we have U ( p ( z + a )) = p ( z ) where p ( z + a ) is also apolynomial. The non-orthonormality is clear since 1 ( z − a ) n for n > a = 0.As an immediate application of Theorem 5.2 we obtain one of the main results of Narayan,Sievewright and Thompson [15]. Corollary 5.3. If φ is an affine self-map with a fixed point in D , then C φ is complex sym-metric on H . Proof.
Let φ ( z ) = cz + d with fixed point a = d − c ∈ D . Then we see that C φ ( z − a ) n = c n ( z − a ) n where | c | < z − a ) n ) n ∈ N is the Koenigs sequence for φ . Therefore C φ is complex symmetric by Theorems 3.1 and 5.2.8 . Commutants of complex symmetric composition operators In this section, we study how the complex symmetry of C φ affects the structure of itscommutant C ′ φ when φ is a Schr¨oder symbol. The main result is the following. Theorem 6.1.
Let φ be a Schr¨oder map such that C φ is complex symmetric on H . Theneach A ∈ C ′ φ is complex symmetric. Proof.
Let a ∈ D be the fixed point of φ , σ its Koenigs eigenfunction and λ = φ ′ ( a ). If A commutes with C φ then C φ Aσ n = λ n Aσ n for each n ∈ N . By Koenigs’ Theorem there mustexist a n ∈ C such that Aσ n = a n σ n for n ∈ N . If C φ is J -symmetric, then for n, m ∈ N wehave h σ m , ( A ∗ − a n I ) J σ n i = h ( A − a n I ) σ m , J σ n i = ( a m − a n ) h σ m , J σ n i . Since ( σ n ) n ∈ N is complete J -orthogonal by Theorem 3.1, we obtain A ∗ J σ n = a n J σ n . Hence A ∗ J σ n = a n J σ n = J ( a n σ n ) = J Aσ n and the completeness of ( σ n ) n ∈ N implies A ∗ J = J A .The conclusion of Theorem 6.1 does not hold in general. Let φ = φ a ◦ ( − φ a ) with a ∈ D \{ } . Then φ is elliptic of order 2 and C φ is complex symmetric. If Φ a,λ = φ a ◦ ( λφ a )where λ ∈ D \{ } , then Φ a,λ commutes with φ and hence C Φ a,λ ∈ C ′ φ even though C Φ a,λ is not complex symmetric by Theorem 4.1. An immediate implication of Theorem 6.1 is thefollowing interesting result. Corollary 6.2.
Let φ be a Schr¨oder map. Then C φ is complex symmetric if and only if C nφ is complex symmetric for some integer n ≥ . Proof. If C φ is complex symmetric then clearly so is C nφ for all n ∈ N . Conversely suppose C nφ is complex symmetric for some n ≥
2. Clearly the n -th composite φ [ n ] of φ is also aSchr¨oder map. Since C φ [ n ] = C nφ is complex symmetric and C φ commutes with C φ [ n ] , itfollows from Theorem 6.1 that C φ is complex symmetric.Again this is not true in general, since if φ is an elliptic automorphism of order 4 then C φ is not complex symmetric (see [1, Prop. 3.3]), but C φ is elliptic of order two and is hencecomplex symmetric. Acknowledgement
This work constitutes a part of the doctoral dissertation of the second author under thesupervision of the first author. 9 eferences [1] P. S. Bourdon, S. W. Noor,
Complex symmetry of invertible composition operators , J. Math. Anal.Appl. 429 (2015), no. 1, 105110.[2] P. S. Bourdon, J. H. Shapiro,
Riesz composition operators , Pacific J. Math. 181 (1997), no. 2, 231246.[3] P. S. Bourdon, J. H. Shapiro,
Mean growth of Koenigs eigenfunctions , J. Amer. Math. Soc. 10 (1997),no. 2, 299325.[4] O. Christensen,
An Introduction to Frames and Riesz Bases , Applied and Numerical Harmonic Analysis,Birkhuser Basel (2003).[5] C. C. Cowen, B. D. MacCluer,
Composition operators on spaces of analytic functions , CRC Press, BocaRaton, 1995.[6] S. R. Garcia, C. Hammond,
Which weighted composition operators are complex symmetric? , Oper.Theory Adv. Appl. 236 (2014) 171-179.[7] S. R. Garcia, M. Putinar,
Complex symmetric operators and applications , Trans. Amer. Math. Soc. 358(3) (2006) 1285-1315 (electronic).[8] S. R. Garcia, M. Putinar,
Complex symmetric operators and applications II , Trans. Amer. Math. Soc.359 (8) (2007) 3913-3931 (electronic).[9] S. R. Garcia, M. Putinar,
Interpolation and complex symmetry , Tohoku Math. J. (2) Volume 60, Number3 (2008), 423-440.[10] S. R. Garcia,
The eigenstructure of complex symmetric operators , Recent advances in matrix andoperator theory, 169183, Oper. Theory Adv. Appl., 179, Birkhuser, Basel, 2008.[11] S. R. Garcia, W. R. Wogen,
Complex symmetric partial isometries , J. Funct. Anal. 257 (4) (2009)1251-1260.[12] S. R. Garcia, W. R. Wogen,
Some new classes of complex symmetric operators , Trans. Amer. Math.Soc. 362 (11) (2010) 6065-6077.[13] G. Koenigs,
Recherches sur les intgrales de certaines quations fonctionnelles , Ann. Sci. cole Norm. Sup.(3) 1 (1884), 341.[14] R. A. Mart´ınez-Avenda˜no, P. Rosenthal,
An Introduction to Operators on the Hardy-Hilbert Space .Graduate Texts in Mathematics, vol. 237, Springer-Verlag, New York, 2007.[15] S. Narayan, D. Sievewright, D.Thompson,
Complex symmetric composition operators on H , J. Math.Anal. Appl. 443 (2016), no. 1, 625630.[16] S. K. Narayan, D. Sievewright, M. Tjani, Complex symmetric composition operators on weighted Hardyspaces , Proc. Amer. Math. Soc. 148 (2020), 2117-2127[17] S. W. Noor.
Complex symmetry of composition operators induced by involutive ball automorphisms .Proc. Amer. Math. Soc. 142 (2014), no. 9, 31033107.[18] S. W. Noor, R. Osmar Severiano,
Complex symmetry and cyclicity of composition operators on H ( C + ) . Proc. Amer. Math. Soc. 148 (2020), no. 6, 24692476.[19] P. Poggi-Corradini,
The Hardy class of Koenigs maps , Michigan Math. J. 44 (1997), no. 3, 495507.[20] Sebastian Ruiz,
An Algebraic Identity Leading to Wilson’s Theorem , The Mathematical Gazette. 80(1996) (489): 579582.[21] J. H. Shapiro,
Composition operators and classical function theory , Tracts in Mathematics. Springer-Verlag, New York, 1993., Tracts in Mathematics. Springer-Verlag, New York, 1993.