Some Notes on Complex Symmetric Operators

In this paper we show that every conjugation $C$ on the Hardy-Hilbert space $H^{2}$ is of type $C=T^{*}C_{1}T$, where $T$ is an unitary operator and $C_{1}f\left(z\right)=\overline{f\left(\overline{z}\right)}$, with $f\in H^{2}$. In the sequence, we extend this result for all separable Hilbert space $\mathcal H$ and we prove some properties of complex symmetry on $\mathcal H$. Finally, we prove some relations of complex symmetry between the operators $T$ and $\left|T\right|$, where $T =U\left|T\right|$ is the polar decomposition of bounded operator $T\in\mathcal L\left(\mathcal H\right)$ on the separable Hilbert space $\mathcal H$.


Introduction
Let L (H) be the space of bounded linear operators on a separable Hilbert space H. A conjugation C on H is an antilinear operator C : H → H such that C 2 = I and Cf, Cg = g, f , for all f, g ∈ H. An operator T ∈ L (H) is said to be complex symmetric if there exists a conjugation C on H such that CT = T * C (we will often say that T is C-symmetric). Complex symmetric operators generalize the concept of symmetric matrices of linear algebra. Indeed, it is well known ([5, Lemma 1]) that given a conjugation C, there exists an orthonormal basis {f n } ∞ n=0 for H such that Cf n = f n . Hence, if T is C-symmetric then that is, T has a symmetric matrix representation. The reciprocal of this fact is also true. That is, if there is an orthonormal basis such that T has a symmetric matrix representation, then T is complex symmetric. The complex symmetric operators class was initially addressed by Garcia and Putinar [5,6] and includes the normal operators, Hankel operators and Volterra integration operators. Now, let L 2 be the Hilbert space on the unit circle T and let L ∞ be the Banach space of all essentially bounded functions on T. It is known that {e n (e it ) := e int : n ∈ Z} is an orthonormal basis for L 2 . The Hardy-Hilbert space, denoted by H 2 , consists of all analytic functions f (z) = ∞ n=0 a n z n on the unit disk D such that ∞ n=0 |a n | 2 < ∞. It is clear that B := {e n (z) = z n : n = 0, 1, 2, . . .} is an orthonormal basis for H 2 .
For each φ ∈ L ∞ , the Toeplitz operator T φ : H 2 → H 2 is defined by for each f ∈ H 2 , where P : L 2 → H 2 is the orthogonal projection. The concept of Toeplitz operators was initiated by Brown and Halmos [1] and generalizes the concept of Toeplitz matrices. In [7], Guo and Zhu raised the question of characterizing complex symmetric Toeplitz operators on H 2 in the unit disk. In order to obtain such characterization, Ko and Lee [8] introduced the family of conjugations C λ : H 2 → H 2 , given by with λ ∈ T and proved the following result:

Canonical Conjugations
Our first objective in this paper is to study relations between an arbitrary conjugation C on H 2 and the conjugation C 1 f (z) = f (z). Once the conjugation C 1 is a kind of canonical conjugation on H 2 , we observe a close relationship between conjugations of H 2 and conjugation C 1 , namely: Theorem 2.1. If C is an conjugation on H 2 , then exists an unitary operator T : H 2 → H 2 such that T C = C 1 T.
Proof. Since C is an conjugation, there exists an orthonormal basis n=0 the standard orthonormal basis of H 2 and the linear isomorphism T : H 2 → H 2 given by T ∞ n=0 a n f n = ∞ n=0 a n e n .
Note that T f n = e n , for all n ≥ 0, and therefore T is unitary. Now, for f (z) = ∞ n=0 a n e n ∈ H 2 , we get The previous theorem says that all complex symmetric Toeplitz operator is unitarily equivalent to a C 1 -symmetric operator. Indeed: therefore the operator T 2 := T T φ T * is C 1 -symmetric (see [5, p. 1291]). This shows that T φ and T 2 are unitarily equivalent operators. Moreover, is obvious that, if T commutes with C 1 or C, then C = C 1 .
Thus, we must that is C 1 e n = e n , ∀n ≥ 0. Therefore, by (1), follows that Reciprocally, suppose that A is C-symmetric such that Ce n = e n . By previous theorem, T C = C 1 T and T e n = e n . Hence, T is the identity operator and so C = C 1 .
In fact, the reciprocal of the Theorem 2.1 is true: Proof. It is easy to see that C is an antilinear operator. Now, since T is an unitary operator, considering B = {e n } ∞ n=0 the orthonormal basis of H 2 , we have We already know that every normal operator is complex symmetric and that the reciprocal in general is not true. However, for Toeplitz operators, Theorem 1.1 gives us: Now note that if T φ is normal not necessarily T φ is J -symmetric. In fact, if φ(z) = −z + z then T φ is normal, however is not J -symmetric.

Properties of Complex Symmetry
In the following, we present some properties of complex symmetry in Hilbert spaces. The first result gives us a way to get complex symmetric operators from another complex symmetric operator. First, we need some lemmas:  Proof. We already know that U is unitary and C and J-symmetric and that UC = CJC is a conjugation, by Lemmas 3.1 and 3.2. Now since U * = U −1 = JC and T is Csymmetric, we have Reciprocally, suppose that UC(UT ) * = UT (UC). Thus and so T is CJC-symmetric. Analogous, we prove that T is JCJ-symmetric. Proof. We already know that U = CJ is unitary and both C and J -symmetric. Now, note that First see that if T is C-symmetric, then UT * C = U(CT ) = (CU * )T . Reciprocally, we have

Complex Symmetry of Aluthge and Duggal Transforms
Recall that the polar decomposition of an operator T : H → H is uniquely expressed by T = U |T |, where |T | = √ T * T is a positive operator and U is a partial isometry such that Ker(U) = Ker |U| and U maps cl(Ran |T |) onto cl(Ran(T )). In this case, the Aluthge and Duggal Transforms are given, respectively, by T = |T | We already known that the Aluthge transform of a complex symmetric operator is also complex symmetric (see [4,Theorem 1]). In this section we study relations between complex symmetry of T and |T | with relation the conjugations C and J, as well as the operators T and T .