Open AccessArticle

2-Complex Symmetric Composition Operators on H²

Lian Hu

Songxiao Li

and

Rong Yang

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China

Author to whom correspondence should be addressed.

Axioms 2022, 11(8), 358; https://doi.org/10.3390/axioms11080358

Submission received: 24 June 2022 / Revised: 10 July 2022 / Accepted: 21 July 2022 / Published: 23 July 2022

(This article belongs to the Special Issue Symmetry of Nonlinear Operators)

Download Versions Notes

Abstract

In this paper, we study 2-complex symmetric composition operators with the conjugation J, defined by

J f (z) = \bar{(f (\bar{z}))}

, on the Hardy space

H^{2}

. More precisely, we obtain the necessary and sufficient condition for the composition operator

C_{ϕ}

to be 2-complex symmetric with J when

ϕ

is an automorphism of

D

. We also characterize 2-complex symmetric with J when

ϕ

is a linear fractional self-map of

D

Keywords:

composition operator; m-complex symmetric; normal

MSC:

30H10; 47B38

1. Introduction

Throughout this paper, H and

B (H)

will always denote a separable complex Hilbert space and the set of all continuous linear operators on H, respectively.

Definition 1.

An operator

C : H \to H

is said to be a conjugation on H if it is

(a): anti-linear or conjugate-linear: $C (α x + β y) = \bar{α} C (x) + \bar{β} C (y)$ , for all $α, β \in C$ and $x, y \in H$ ,
(b): isometric: $∥ C x ∥ = ∥ x ∥$ , for all $x \in H$ ,
(c): involutive: $C^{2} = I_{d}$ , where $I_{d}$ is an identity operator.

It is easy to check that

(J f) (z) = \bar{f (\bar{z})}

is a conjugation on the Hardy space

H^{2}

Definition 2.

For a conjugation C on H, an operator

T \in B (H)

is said to be complex symmetric (complex symmetric with C or a C-symmetric operator) if

T = C T^{*} C

The class of complex symmetric operators includes all normal operators, binormal operators, Hankel operators, compressed Toeplitz operators and Volterra integration operators. The study of complex symmetric operators was initiated by Garcia, Putinar, and Wogen in [1,2,3,4]. See [5,6,7,8,9,10,11,12,13,14] for more results on complex symmetric operators.

Definition 3.

Let m be a positive integer and

T \in B (H)

. T is said to be a m-complex symmetric operator (m-complex symmetric with C) if there exists a conjugation C such that

\sum_{j = 0}^{m} {(- 1)}^{m - j} (\begin{matrix} m \\ j \end{matrix}) {T *}^{j} C T^{m - j} C = 0 .

The above definition was introduced by Chō, Ko, and Lee in [15]. When

m = 2

, we obtain

C T^{2} C - 2 T^{*} C T C + {T^{*}}^{2} = 0,

which is equivalent to

C T^{2} - 2 T^{*} C T + {T^{*}}^{2} C = 0 .

It is clear that 1-complex symmetric operator is just the complex symmetric operator. From [15], we see that all complex symmetric operators are 2-complex symmetric operators. Thus, the set of all 2-complex symmetric operators is larger than the set of all complex symmetric operators. In [16], the authors studied m-complex symmetric weighted shifts on

C^{n}

. We refer the reader to [15,16,17,18] for more results about m-complex symmetric operators.

Inspired by these papers, in this paper, we study 2-complex symmetric composition operators, induced by linear fractional self-maps of

D

, with J on the Hardy space

H^{2}

. When the symbol

ϕ

is an automorphism of

D

, we show that the composition operator

C_{ϕ}

is 2-complex symmetric with J if and only if

C_{ϕ}

is normal. Furthermore, we also characterize 2-complex symmetric composition operators with J on

H^{2}

when the induced maps are linear fractional self-maps of

D

2. Preliminaries

Let

D

and

\partial D

be the open unit disk

C

and the unit circle in the complex plane, respectively. Let

H (D)

be the set of all analytic functions on

D

. The Hardy space

H^{2} (D)

is the space of all

f \in H (D)

such that

{∥ f ∥}^{2} = sup_{0 \leq r < 1} \frac{1}{2 π} \int_{0}^{2 π} {| f (r e^{i θ}) |}^{2} d θ < \infty .

The space

H^{2} (D)

is a reproducing kernel Hilbert space, that is, for each

w \in D

and

f \in H^{2} (D)

, there is a unique function

K_{w}

such that

〈 f, K_{w} 〉 = f (w),

where

K_{w} (z) = \frac{1}{1 - \bar{w} z}

is said to be the reproducing kernel at w. For

g \in L^{\infty} (\partial D)

, the Toeplitz operator

T_{g}

is defined as

T_{g} f = P (f g)

for

f \in H^{2} (D)

, where P is the orthogonal projection of

L^{2}

onto

H^{2} (D)

. Recall that

T_{g}^{*} K_{w} = \bar{f (w)} K_{w}

for each

w \in D

and

f \in H^{\infty}

Let

ϕ

be an analytic self-map of

D

. Recall that the composition operator

C_{ϕ}

is defined by

C_{ϕ} f (z) = f (ϕ (z)), f \in H (D), z \in D .

It is easy to see that

C_{ϕ}^{*} K_{w} = K_{ϕ (w)}

for each

w \in D

3. Main Results

We begin this section with Cowen’s formula for the adjoint of a linear fractional self-map. For a linear fractional self-map

ϕ (z) = \frac{a z + b}{c z + d}

, Cowen in [19] obtained the following important formula:

C_{ϕ}^{*} = T_{g} C_{σ} T_{h}^{*},

where

a, b, c, d \in C

a d - b c \neq 0

and

σ (z) = \frac{\bar{a} z - \bar{c}}{- \bar{b} z + \bar{d}}, g (z) = \frac{1}{- \bar{b} z + \bar{d}}, h (z) = c z + d .

Next, we state some lemmas which will be used in our mian results.

Lemma 1

([20]). Let ϕ be an analytic self-map of

D

. Then,

C_{ϕ}

is normal if and only if

ϕ (z) = a z

with

| a | \leq 1

Lemma 2.

Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-map of

D

. If

w \in D

satisfies

\bar{a} w \neq \bar{c}

, then the following statements hold:

(a): $J C_{ϕ}^{2} K_{w} (z) = K_{\bar{w}} (\bar{ϕ_{2} (\bar{z})})$ ,
(b): $C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = - \bar{c} \frac{g (w)}{σ (w)} K_{ϕ (0)} (z) + \bar{h (\frac{1}{\bar{σ (w)}})} g (w) K_{ϕ (\bar{σ (w)})} (z),$
(c): ${C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z),$
(d): $J C_{ϕ} J C_{ϕ}^{*} J K_{w} (z) = K_{ϕ (\bar{w})} (\bar{ϕ (\bar{z})}),$

where

σ (z) = \frac{\bar{a} z - \bar{c}}{- \bar{b} z + \bar{d}}

g (z) = \frac{1}{- \bar{b} z + \bar{d}}

and

h (z) = c z + d

Proof.

Since

J f (z) = \bar{f (\bar{z})}

and

f (w) = 〈 f, K_{w} 〉

for any

w \in D

and

f \in H^{2}

, we see that

(a)

and

(c)

hold obviously. Now, we only verify

(b)

and

(d)

. For any

w \in D

with

\bar{a} w \neq \bar{c}

, we obtain

\begin{matrix} C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = & C_{ϕ}^{*} K_{\bar{w}} (\bar{ϕ (\bar{z})}) \\ = & C_{ϕ}^{*} \frac{\bar{c} z + \bar{d}}{(\bar{c} - \bar{a} w) z + (\bar{d} - \bar{b} w)} \\ = & C_{ϕ}^{*} (\frac{\bar{c}}{\bar{c} - \bar{a} w} + (\bar{d} - \frac{\bar{c} (\bar{d} - \bar{b} w)}{\bar{c} - \bar{a} w}) \frac{1}{\bar{d} - \bar{b} w} \frac{1}{1 - \frac{\bar{a} w - \bar{c}}{\bar{d} - \bar{b} w} z}) \\ = & \frac{\bar{c}}{\bar{c} - \bar{a} w} C_{ϕ}^{*} K_{0} (z) + (\bar{d} - \frac{\bar{c} (\bar{d} - \bar{b} w)}{\bar{c} - \bar{a} w}) \frac{1}{\bar{d} - \bar{b} w} C_{ϕ}^{*} K_{\bar{σ (w)}} (z) \\ = & - \bar{c} \frac{g (w)}{σ (w)} K_{ϕ (0)} (z) + \bar{h (\frac{1}{\bar{σ (w)}})} g (w) K_{ϕ (\bar{σ (w)})} (z) . \end{matrix}

Clearly,

\begin{matrix} J C_{ϕ} J C_{ϕ}^{*} J K_{w} (z) & = J C_{ϕ} J C_{ϕ}^{*} K_{\bar{w}} (z) = J C_{ϕ} J K_{ϕ (\bar{w})} (z) \\ = J C_{ϕ} K_{\bar{ϕ (\bar{w})}} (z) = J K_{\bar{ϕ (\bar{w})}} (ϕ (z)) \\ = K_{ϕ (\bar{w})} (\bar{ϕ (\bar{z})}) . \end{matrix}

The proof is complete. □

Lemma 3.

Let ϕ be an analytic self-map of

D

. If

C_{ϕ}

is normal, then

C_{ϕ}^{*}

and

C_{ϕ}

is 2-complex symmetric with J.

Proof.

Assume that

C_{ϕ}

is normal. From Corollary 3.10 in [21], we see that

C_{ϕ}

is complex symmetric with J. Hence,

C_{ϕ}

is 2-complex symmetric with J. Write

ϕ (z) = a z

with

| a | \leq 1

from Lemma 1. We notice that

T^{*}

is 2-complex symmetric with J if and only if

{T^{*}}^{2} J - 2 J T J T^{*} J + J T^{2} = 0 .

In view of Lemma 2, set

σ (z) = \bar{a} z

g (z) = h (z) = 1

, we have

{C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - {\bar{a}}^{2} w z},

\begin{matrix} J C_{ϕ} J C_{ϕ}^{*} J K_{w} (z) & = K_{ϕ (\bar{w})} (\bar{ϕ (\bar{z})}) = \frac{1}{1 - {\bar{a}}^{2} w z} \end{matrix}

and

J C_{ϕ}^{2} K_{w} (z) = K_{\bar{w}} (\bar{ϕ_{2} (\bar{z})}) = \frac{1}{1 - w \bar{ϕ_{2} (\bar{z})}} = \frac{1}{1 - {\bar{a}}^{2} w z} .

Thus, we obtain that

{C_{ϕ}^{*}}^{2} J - 2 J C_{ϕ} J C_{ϕ}^{*} J + J C_{ϕ}^{2} = 0

, which means that

C_{ϕ}^{*}

is 2-complex symmetric with J. □

Lemma 4.

Let ϕ be an analytic self-map of

D

. If

C_{ϕ}

is 2-complex symmetric with J, then

2 ϕ_{2} (0) - 4 ϕ (0) + ϕ (0) ϕ_{2} (0) = 0 .

Proof.

Since

f (w) = 〈 f, K_{w} 〉

for any

w \in D

and

f \in H^{2}

, we have

〈 J C_{ϕ}^{2} K_{0}, K_{\frac{1}{2}} 〉 = 〈 J K_{0}, K_{\frac{1}{2}} 〉 = 1,

〈 C_{ϕ}^{*} J C_{ϕ} K_{0}, K_{\frac{1}{2}} 〉 = 〈 C_{ϕ}^{*} K_{0}, K_{\frac{1}{2}} 〉 = 〈 K_{ϕ (0)}, K_{\frac{1}{2}} 〉 = \frac{2}{2 - \bar{ϕ (0)}}

and

〈 {C_{ϕ}^{*}}^{2} J K_{0}, K_{\frac{1}{2}} 〉 = 〈 {C_{ϕ}^{*}}^{2} K_{0}, K_{\frac{1}{2}} 〉 = 〈 K_{ϕ_{2} (0)}, K_{\frac{1}{2}} 〉 = \frac{2}{2 - \bar{ϕ_{2} (0)}} .

By the assumption that

C_{ϕ}

is 2-complex symmetric with J, we obtain that

1 - \frac{4}{2 - \bar{ϕ (0)}} + \frac{2}{2 - \bar{ϕ_{2} (0)}} = 0 .

By a simple calculation, we obtain the desired result. □

The following result gives a necessary and sufficient condition for

C_{ϕ}

to be 2-complex symmetric with J when

ϕ

is an automorphism of

D

Theorem 1.

Let ϕ be an automorphism of

D

. Then,

C_{ϕ}

is 2-complex symmetric with J if and only if

C_{ϕ}

is normal.

Proof.

Assume that

C_{ϕ}

is normal. From Lemma 3, we see that

C_{ϕ}

is 2-complex symmetric with J.

Now, suppose that

ϕ (z) = λ \frac{a - z}{1 - \bar{a} z}

with

| λ | = 1

a \in D

and

C_{ϕ}

is 2-complex symmetric with J. Then,

σ (z) = \frac{- \bar{λ} z + a}{- \bar{λ a} z + 1}

g (z) = \frac{1}{- \bar{λ a} z + 1}

and

h (z) = - \bar{a} z + 1

. Noting that

ϕ_{2} (z) = \frac{λ (λ z - λ a) - λ a (\bar{a} z - 1)}{\bar{a} (λ z - λ a) - (\bar{a} z - 1)},

then, for

\bar{λ} w \neq a

, Lemma 2 gives that

\begin{matrix} J C_{ϕ}^{2} K_{w} (z) = K_{\bar{w}} (\bar{ϕ_{2} (\bar{z})}) = \frac{a (\bar{λ} z - \bar{λ a}) - (a z - 1)}{a (\bar{λ} z - \bar{λ a}) - (a z - 1) - w [\bar{λ} (\bar{λ} z - \bar{λ a}) - \bar{λ a} (a z - 1)]}, \end{matrix}

(1)

\begin{matrix} C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = & a \frac{g (w)}{σ (w)} K_{ϕ (0)} (z) + \bar{h (\frac{1}{\bar{σ (w)}})} g (w) K_{ϕ (\bar{σ (w)})} (z) \\ = & \frac{a}{(- \bar{λ} w + a) (1 - \bar{λ a} z)} + \frac{\bar{λ} {w (| a |}^{2} - 1)}{(- \bar{λ} w + a) (- \bar{λ a} w + 1)} \\ \cdot \frac{- \bar{λ a} w + 1 + a \bar{λ} w - a^{2}}{- \bar{λ a} w + 1 + a \bar{λ} w - a^{2} - ({\bar{λ}}^{2} w - {\bar{λ}}^{2} {\bar{a}}^{2} w - \bar{λ} a + \bar{λ a}) z} \end{matrix}

(2)

and

\begin{matrix} {C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{a (\bar{λ} w - \bar{λ a}) - (a w - 1)}{a (\bar{λ} w - \bar{λ a}) - (a w - 1) - [\bar{λ} (\bar{λ} w - \bar{λ a}) - \bar{λ a} (a w - 1)] z} \end{matrix}

(3)

for any

w, z \in D

. Taking

w = 0

in (1)–(3), we have

J C_{ϕ}^{2} K_{0} (z) = 1, C_{ϕ}^{*} J C_{ϕ} K_{0} (z) = \frac{1}{1 - \bar{λ a} z},

and

{C_{ϕ}^{*}}^{2} J K_{0} (z) = \frac{- \bar{λ} {| a |}^{2} + 1}{- \bar{λ} {| a |}^{2} + 1 + ({\bar{λ}}^{2} \bar{a} - \bar{λ a}) z}

for any

z \in D

. Since

C_{ϕ}

is 2-complex symmetric with J, we obtain

\begin{matrix} \frac{1 + \bar{λ a} z}{1 - \bar{λ a} z} = \frac{1 - \bar{λ} {| a |}^{2}}{- \bar{λ} {| a |}^{2} + 1 + ({\bar{λ}}^{2} \bar{a} - \bar{λ a}) z} . \end{matrix}

(4)

Calculating and noting that the coefficient of

z^{2}

must be 0, we obtain that

λ^{2} a^{2} (λ - 1) = 0

. Since

λ

and a are non-zero, then

λ = 1

. Hence, (4) becomes

\frac{1 + \bar{a} z}{1 - \bar{a} z} = 1,

which implies that

\bar{a} z = 0

for all

z \in D

. Thus,

a = 0

. Hence,

ϕ (z) = - λ z

with

| λ | = 1

. Lemma 1 gives that

C_{ϕ}

is normal. The proof is complete. □

4. Linear Fractional Self-Maps

In this section, we first consider 2-complex symmetric composition operators with J which are induced linear fractional self-maps with

ϕ (0) = 0

. In the following, we obtain a sufficient and necessary condition for this case.

Theorem 2.

Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-map of

D

and

ϕ (0) = 0

. Then,

C_{ϕ}

(

C_{ϕ}^{*}

) is 2-complex symmetric with J if and only if

C_{ϕ}

is normal.

Proof.

Assume first that

C_{ϕ}

is normal. Lemma 3 gives that

C_{ϕ}

(

C_{ϕ}^{*}

) is 2-complex symmetric with J.

Conversely, suppose that

C_{ϕ}

is 2-complex symmetric with J. Since

a \neq 0

and

ϕ (0) = 0

, set

ϕ (z) = \frac{z}{s z + t}

, where

s = \frac{c}{a}

and

t = \frac{d}{a}

satisfy

| t | \geq | s | + 1

. Then,

σ (z) = \frac{z - \bar{s}}{\bar{t}}

g (z) = \frac{1}{\bar{t}}

and

h (z) = s z + t

. If

s = 0

, then

ϕ (z) = \frac{1}{t} z

. Therefore, Lemma 1 gives that

C_{ϕ}

is normal. Now, we suppose that

s \neq 0

. After a calculation, we have that

ϕ_{2} (z) = \frac{z}{(s + s t) z + t^{2}}

and

ϕ (\bar{σ (z)}) = \frac{\bar{z} - s}{s \bar{z} - s^{2} + t^{2}}, h (\frac{1}{\bar{σ (z)}}) = \frac{s t}{\bar{z} - s} + t .

For any

w, z \in D

with

w \neq \bar{s}

, employing Lemma 2, we obtain that

\begin{matrix} J C_{ϕ}^{2} K_{w} (z) = K_{\bar{w}} (\bar{ϕ_{2} (\bar{z})}) = \frac{1}{1 - \frac{z w}{\bar{s} z + \bar{s t} z + {\bar{t}}^{2}}} = \frac{(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}}{(\bar{s} + \bar{s t}) z + {\bar{t}}^{2} - w z}, \end{matrix}

(5)

\begin{matrix} C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = & - \bar{s} \frac{g (w)}{σ (w)} K_{ϕ (0)} (z) + \bar{h (\frac{1}{\bar{σ (w)}})} g (w) K_{ϕ (\bar{σ (w)})} (z) \\ = & \frac{- \bar{s}}{w - \bar{s}} + (\frac{\bar{s t}}{w - \bar{s}} + \bar{t}) \cdot \frac{1}{\bar{t}} \frac{1}{1 - \frac{w - \bar{s}}{\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2}} z} \\ = & \frac{\bar{s}}{\bar{s} - w} + \frac{w \bar{t}}{(w - \bar{s}) \bar{t}} \frac{\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2}}{\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2} - (w - \bar{s}) z}, \end{matrix}

(6)

and

\begin{matrix} {C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - \frac{z w}{\bar{s} w + \bar{s t} w + {\bar{t}}^{2}}} = \frac{(\bar{s} + \bar{s t}) w + {\bar{t}}^{2}}{(\bar{s} + \bar{s t}) w + {\bar{t}}^{2} - w z} . \end{matrix}

(7)

Since

C_{ϕ}

is 2-complex symmetric with J, we obtain from (5)–(7) that

\frac{(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}}{(\bar{s} + \bar{s t} - w) z + {\bar{t}}^{2}} + \frac{(\bar{s} + \bar{s t}) w + {\bar{t}}^{2}}{(\bar{s} + \bar{s t}) w + {\bar{t}}^{2} - w z}

= \frac{2 \bar{s}}{\bar{s} - w} + \frac{2 w \bar{t}}{(w - \bar{s}) \bar{t}} \frac{\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2}}{\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2} - (w - \bar{s}) z}

for any

w, z \in D

with

w \neq \bar{s}

, which gives that

\begin{matrix} \frac{[(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}] [(\bar{s} + \bar{s t}) w + {\bar{t}}^{2} - w z] + [(\bar{s} + \bar{s t}) w + {\bar{t}}^{2}] [(\bar{s} + \bar{s t} - w) z + {\bar{t}}^{2}]}{[(\bar{s} + \bar{s t} - w) z + {\bar{t}}^{2}] [(\bar{s} + \bar{s t}) w + {\bar{t}}^{2} - w z]} \\ = & \frac{2 \bar{s t} (w - \bar{s}) [\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2} - (w - \bar{s}) z] + 2 w \bar{t} (\bar{s} - w) (\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2})}{- {(w - \bar{s})}^{2} \bar{t} [\bar{s} w - {\bar{s}}^{2} + {\bar{t}}^{2} - (w - \bar{s}) z]} \end{matrix}

(8)

for any

w, z \in D

with

w \neq \bar{s}

. Taking

w = 0

in (8), then

\frac{{\bar{t}}^{2} [(\bar{s} - \bar{s t}) z + {\bar{t}}^{2}] + {\bar{t}}^{2} [(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}]}{{\bar{t}}^{2} [(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}]} = \frac{2 {\bar{s}}^{2} \bar{t} (- {\bar{s}}^{2} + {\bar{t}}^{2} + \bar{s} z)}{{\bar{s}}^{2} \bar{t} (- {\bar{s}}^{2} + {\bar{t}}^{2} + \bar{s} z)} = 2 .

Therefore,

(\bar{s} + \bar{s t}) z + {\bar{t}}^{2} = (\bar{s} - \bar{s t}) z + {\bar{t}}^{2}

for all

z \in D

. Since

s \neq 0

, then

1 + t = 1 - t

. Hence,

t = 0

, a contradiction. Thus, the hypothesis is not true that is

s = 0

Now, assume that

C_{ϕ}^{*}

is 2-complex symmetric with J. We also assume that

s \neq 0

. By Lemma 2, we have

\begin{matrix} J C_{ϕ} J C_{ϕ}^{*} J K_{w} (z) & = K_{ϕ (\bar{w})} (\bar{ϕ (\bar{z})}) = \frac{1}{1 - \frac{w}{\bar{s} w + \bar{t}} \cdot \frac{z}{\bar{s} z + \bar{t}}} \\ = \frac{(\bar{s} w + \bar{t}) (\bar{s} z + \bar{t})}{(\bar{s} w + \bar{t}) (\bar{s} z + \bar{t}) - w z} \end{matrix}

(9)

for any

w, z \in D

. Since

C_{ϕ}^{*}

is 2-complex symmetric with J, we obtain from (5), (7), and (9) that

\begin{matrix} \frac{[(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}] [(\bar{s} + \bar{s t}) w + {\bar{t}}^{2} - w z] + [(\bar{s} + \bar{s t}) w + {\bar{t}}^{2}] [(\bar{s} + \bar{s t} - w) z + {\bar{t}}^{2}]}{[(\bar{s} + \bar{s t} - w) z + {\bar{t}}^{2}] [(\bar{s} + \bar{s t}) w + {\bar{t}}^{2} - w z]} \\ = & \frac{2 (\bar{s} w + \bar{t}) (\bar{s} z + \bar{t})}{(\bar{s} w + \bar{t}) (\bar{s} z + \bar{t}) - w z} \end{matrix}

(10)

for any

w, z \in D

with

w \neq \bar{s}

. Taking

w = 0

in (10), we also have that

\frac{{\bar{t}}^{2} [(\bar{s} - \bar{s t}) z + {\bar{t}}^{2}] + {\bar{t}}^{2} [(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}]}{{\bar{t}}^{2} [(\bar{s} + \bar{s t}) z + {\bar{t}}^{2}]} = 2 .

The other arguments are similar to the case of

C_{ϕ}

. Then, we obtain the desired result. The proof is complete. □

We now consider 2-complex symmetric composition operators with J which are induced linear fractional self-maps with

c = 0

Lemma 5

([22]). Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-mapof

D

such that

c = 0

. Then,

C_{ϕ} = C_{ξ}^{*} T_{τ}^{*}

, where

ξ (z) = \frac{\bar{a} z}{- \bar{b} z + \bar{d}}

and

τ (z) = \frac{\bar{d}}{- \bar{b} z + \bar{d}}

In the next two results, we give the necessary and sufficient conditions for

C_{ϕ}

and its adjoint operator

C_{ϕ}^{*}

to be 2-complex symmetric with J when

c = 0

Theorem 3.

Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-map of

D

such that

c = 0

. Then,

C_{ϕ}

is 2-complex symmetric with J if and only if

C_{ϕ}

is normal.

Proof.

Assume first that

C_{ϕ}

is normal. Lemma 3 gives that

C_{ϕ}

is 2-complex symmetric with J.

Conversely, suppose that

C_{ϕ}

is 2-complex symmetric with J. Since

c = 0

, then

ϕ (z) = m z + n

, where

m = \frac{a}{d}

and

n = \frac{b}{d}

satisfy

| m | + | n | \leq 1

. Let

ξ (z) = \frac{\bar{m} z}{- \bar{n} z + 1}

and

τ (z) = \frac{1}{- \bar{n} z + 1}

. Note that

ϕ_{2} (z) = m^{2} z + m n + n

and

ξ_{2} (z) = \frac{{\bar{m}}^{2} z}{- \bar{m n} z - \bar{n} z + 1}, ϕ (\bar{ξ (z)}) = \frac{m^{2} \bar{z} - n^{2} \bar{z} + n}{- n \bar{z} + 1}, τ (ξ (z)) = \frac{- \bar{n} z + 1}{- \bar{m n} z - \bar{n} z + 1} .

Lemmas 2 and 5 give that

\begin{matrix} J C_{ϕ}^{2} K_{w} (z) & = J C_{ξ}^{*} T_{τ}^{*} C_{ξ}^{*} T_{τ}^{*} K_{w} (z) = J \bar{τ (w)} C_{ξ}^{*} T_{τ}^{*} K_{ξ (w)} (z) \\ = J \bar{τ (w) τ (ξ (w))} K_{ξ_{2} (w)} (z) = τ (w) τ (ξ (w)) K_{\bar{ξ_{2} (w)}} (z) \\ = \frac{1}{- \bar{n} w + 1} \frac{- \bar{n} w + 1}{- \bar{m n} w - \bar{n} w + 1} \cdot \frac{1}{1 - \frac{{\bar{m}}^{2} w}{- \bar{m n} w - \bar{n} w + 1} z} \\ = \frac{1}{- \bar{m n} w - \bar{n} w + 1 - {\bar{m}}^{2} w z}, \end{matrix}

(11)

\begin{matrix} C_{ϕ}^{*} J C_{ϕ} K_{w} (z) & = C_{ϕ}^{*} J C_{ξ}^{*} T_{τ}^{*} K_{w} (z) = C_{ϕ}^{*} J \bar{τ (w)} K_{ξ (w)} (z) = C_{ϕ}^{*} τ (w) \bar{K_{ξ (w)} (\bar{z})} \\ = C_{ϕ}^{*} τ (w) K_{\bar{ξ (w)}} (z) = τ (w) K_{ϕ (\bar{ξ (w)})} (z) \\ = \frac{1}{- \bar{n} w + 1} \cdot \frac{1}{1 - \frac{{\bar{m}}^{2} w - {\bar{n}}^{2} w + \bar{n}}{- \bar{n} w + 1} z} \\ = \frac{1}{- \bar{n} w + 1 - ({\bar{m}}^{2} w - {\bar{n}}^{2} w + \bar{n}) z} \end{matrix}

(12)

and

\begin{matrix} {C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - ({\bar{m}}^{2} w + \bar{m n} + \bar{n}) z} \end{matrix}

(13)

for any

w, z \in D

. Since

C_{ϕ}

is 2-complex symmetric with J, then we obtain from (11)–(13) that

\begin{matrix} \frac{1}{- \bar{m n} w - \bar{n} w + 1 - {\bar{m}}^{2} w z} + \frac{1}{1 - ({\bar{m}}^{2} w + \bar{m n} + \bar{n}) z} \\ = & \frac{2}{- \bar{n} w + 1 - ({\bar{m}}^{2} w - {\bar{n}}^{2} w + \bar{n}) z} \end{matrix}

(14)

for any

w, z \in D

. Taking

w = 0

in (14), we have that

1 + \frac{1}{1 - (\bar{m n} + \bar{n}) z} = \frac{2}{1 - \bar{n} z}

for any

z \in D

. Then,

\frac{2 - (\bar{m n} + \bar{n}) z}{1 - (\bar{m n} + \bar{n}) z} = \frac{2}{1 - \bar{n} z} .

Noting that the coefficient of

z^{2}

must be 0, then

n^{2} (m + 1) = 0

, which means that

n = 0

m = - 1

. Similarly, noting that the coefficient of z must be also 0, then

n = m n

, which means that

n = 0

m = 1

. Therefore,

n = 0

. This implies that

ϕ (z) = m z

with

| m | \leq 1

. Lemma 1 gives that

C_{ϕ}

is normal. □

Theorem 4.

Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-map of

D

such that

c = 0

. Then,

C_{ϕ}^{*}

is 2-complex symmetric with J if and only if

C_{ϕ}

is normal.

Proof.

Since

c = 0

, we can set

ϕ (z) = m z + n

where

m = \frac{a}{d}

and

n = \frac{b}{d}

satisfy

| m | + | n | \leq 1

. Let

ξ (z) = \frac{\bar{m} z}{- \bar{n} z + 1}

and

τ (z) = \frac{1}{- \bar{n} z + 1}

. Noting that

ϕ_{2} (z) = m^{2} z + m n + n,

and

τ (\bar{ϕ (\bar{z})}) = \frac{1}{- \bar{n} (\bar{m} z + \bar{n}) + 1}, ξ (\bar{ϕ (\bar{z})}) = \frac{\bar{m} (\bar{m} z + \bar{n})}{- \bar{n} (\bar{m} z + \bar{n}) + 1} .

Using Lemma 5 and Theorem 3, we obtain that

\begin{matrix} J C_{ϕ}^{2} K_{w} (z) = J C_{ξ}^{*} T_{τ}^{*} C_{ξ}^{*} T_{τ}^{*} K_{w} (z) & = \frac{1}{- \bar{m n} w - \bar{n} w + 1 - {\bar{m}}^{2} w z}, \end{matrix}

(15)

\begin{matrix} J C_{ϕ} J C_{ϕ}^{*} J K_{w} (z) & = J C_{ξ}^{*} T_{τ}^{*} K_{\bar{ϕ (\bar{w})}} (z) \\ = J \bar{τ (\bar{ϕ (\bar{w})})} K_{ξ (\bar{ϕ (\bar{w})})} (z) = τ (\bar{ϕ (\bar{w})}) K_{\bar{ξ (\bar{ϕ (\bar{w})})}} (z) \\ = \frac{1}{- \bar{n} (\bar{m} w + \bar{n}) + 1} \cdot \frac{1}{1 - \frac{\bar{m} (\bar{m} w + \bar{n}) z}{- \bar{n} (\bar{m} w + \bar{n}) + 1}} \\ = \frac{1}{1 - (\bar{m} w + \bar{n}) (\bar{m} z + \bar{n})} \end{matrix}

(16)

and

\begin{matrix} {C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - ({\bar{m}}^{2} w + \bar{m n} + \bar{n}) z} \end{matrix}

(17)

for any

w, z \in D

. Since

C_{ϕ}^{*}

is 2-complex symmetric with J, then we obtain from (15)–(17) that

\frac{1}{- \bar{m n} w - \bar{n} w + 1 - {\bar{m}}^{2} w z} + \frac{1}{1 - ({\bar{m}}^{2} w + \bar{m n} + \bar{n}) z} = \frac{2}{1 - (\bar{m} w + \bar{n}) (\bar{m} z + \bar{n})}

for any

w, z \in D

. Taking

w = 0

, we get

\frac{2 - (\bar{m n} + \bar{n}) z}{1 - (\bar{m n} + \bar{n}) z} = \frac{2}{1 - \bar{n} (\bar{m} z + \bar{n})} .

Noting that the coefficient of constant term must be 0, then

{\bar{n}}^{2} = 0

, which means that

n = 0

. Thus,

ϕ (z) = m z

with

| m | \leq 1

. Lemma 1 assures that

C_{ϕ}

is normal.

Conversely, assume that

C_{ϕ}

is normal. Lemma 3 gives that

C_{ϕ}^{*}

is 2-complex symmetric with J. □

Furthermore, we prove that there is no 2-complex symmetric composition operators with J which are induced linear fractional self-maps with

a = 0

and

ϕ (0) \neq 0

Lemma 6.

Let

ϕ : D \to D

be a constant function. Then,

C_{ϕ}

(C_{ϕ}^{*})

is 2-complex symmetric with J if and only if

ϕ (z) \equiv 0

Proof.

The sufficiency is obvious.

Now, we assume that

ϕ (z) \equiv c

for some

c \in D

and

C_{ϕ}

is 2-complex symmetric with J. Then, we obtain that

J C_{ϕ}^{2} K_{w} (z) = J K_{w} (ϕ_{2} (z)) = J \frac{1}{1 - \bar{w} c} = \frac{1}{1 - w \bar{c}},

C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = C_{ϕ}^{*} J K_{w} (ϕ (z)) = \frac{1}{1 - w \bar{c}} C_{ϕ}^{*} K_{0} (z) = \frac{1}{(1 - w \bar{c}) (1 - \bar{c} z)},

{C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - \bar{c} z}

and

J C_{ϕ} J C_{ϕ}^{*} J K_{w} (z) = K_{ϕ (\bar{w})} (\bar{ϕ (\bar{z})}) = \frac{1}{1 - {\bar{c}}^{2}}

for any

w, z \in D

. Since

C_{ϕ}

is 2-complex symmetric with J, we obtain

\frac{1}{1 - w \bar{c}} - \frac{2}{(1 - w \bar{c}) (1 - \bar{c} z)} + \frac{1}{1 - \bar{c} z} = 0

for any

w, z \in D

. By a simple calculation, we see that

c = 0

Assume that

C_{ϕ}^{*}

is 2-complex symmetric with J. Similarly, we have

\frac{1}{1 - w \bar{c}} - \frac{2}{1 - {\bar{c}}^{2}} + \frac{1}{1 - \bar{c} z} = 0

for any

w, z \in D

. Therefore,

c = 0

. The proof is complete. □

Theorem 5.

Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-map of

D

such that

a = 0

and

ϕ (0) \neq 0

. Then,

C_{ϕ}

is not 2-complex symmetric with J.

Proof.

Since

a = 0

and

ϕ (0) \neq 0

, we set

ϕ (z) = \frac{1}{m z + n}

, where

m = \frac{c}{b}

and

n = \frac{d}{b}

. Then,

σ (z) = \frac{- \bar{m}}{- z + \bar{n}}

g (z) = \frac{1}{- z + \bar{n}}

and

h (z) = m z + n

. When

m = 0

ϕ (z) = \frac{1}{n} \neq 0

, then Lemma 6 gives that

C_{ϕ}

is not 2-complex symmetric with J. Now, we assume that

m \neq 0

n \neq 0

and

C_{ϕ}

is 2-complex symmetric with J. Note that

ϕ_{2} (z) = \frac{m z + n}{m + m n z + n^{2}}, h (\frac{1}{\bar{σ (z)}}) = \bar{z}, ϕ (\bar{σ (z)}) = \frac{- \bar{z} + n}{- m^{2} - n \bar{z} + n^{2}} .

Lemma 2 gives that

\begin{matrix} J C_{ϕ}^{2} K_{w} (z) = K_{\bar{w}} (\bar{ϕ_{2} (\bar{z})}) = \frac{1}{1 - w \frac{\bar{m} z + \bar{n}}{\bar{m} + \bar{m n} z + {\bar{n}}^{2}}} = \frac{\bar{m} + \bar{m n} z + {\bar{n}}^{2}}{\bar{m} + \bar{m n} z + {\bar{n}}^{2} - (\bar{m} z + \bar{n}) w}, \end{matrix}

(18)

\begin{matrix} C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = & - \bar{m} \frac{g (w)}{σ (w)} K_{ϕ (0)} (z) + \bar{h (\frac{1}{\bar{σ (w)}})} g (w) K_{ϕ (\bar{σ (w)})} (z) \\ = & K_{ϕ (0)} (z) + \frac{w}{- w + \bar{n}} K_{ϕ (\bar{σ (w)})} (z) \\ = & \frac{1}{1 - \frac{z}{\bar{n}}} + \frac{w}{- w + \bar{n}} \cdot \frac{1}{1 - \frac{- w + \bar{n}}{- {\bar{m}}^{2} - \bar{n} w + {\bar{n}}^{2}} z} \\ = & \frac{\bar{n}}{\bar{n} - z} + \frac{w}{- w + \bar{n}} \cdot \frac{- {\bar{m}}^{2} - \bar{n} w + {\bar{n}}^{2}}{- {\bar{m}}^{2} - \bar{n} w + {\bar{n}}^{2} - (- w + \bar{n}) z} \end{matrix}

(19)

and

\begin{matrix} {C_{ϕ}^{*}}^{2} J K_{w} (z) = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - z \frac{\bar{m} w + n}{\bar{m} + \bar{m n} w + {\bar{n}}^{2}}} = \frac{\bar{m} + \bar{m n} w + {\bar{n}}^{2}}{\bar{m} + \bar{m n} w + {\bar{n}}^{2} - (\bar{m} w + \bar{n}) z} \end{matrix}

(20)

for any

w, z \in D

. Since

C_{ϕ}

is 2-complex symmetric with J, we obtain from (18)–(20) that

\begin{matrix} \frac{\bar{m} + \bar{m n} z + {\bar{n}}^{2}}{\bar{m} + \bar{m n} z + {\bar{n}}^{2} - (\bar{m} z + \bar{n}) w} + \frac{\bar{m} + \bar{m n} w + {\bar{n}}^{2}}{\bar{m} + \bar{m n} w + {\bar{n}}^{2} - (\bar{m} w + n) z} \\ = & 2 (\frac{\bar{n}}{\bar{n} - z} + \frac{w}{- w + \bar{n}} \cdot \frac{- {\bar{m}}^{2} - \bar{n} w + {\bar{n}}^{2}}{- {\bar{m}}^{2} - \bar{n} w + {\bar{n}}^{2} - (- w + \bar{n}) z}) \end{matrix}

(21)

for any

w, z \in D

. Taking

w = 0

in (21), we have that

\begin{matrix} \frac{(- \bar{n} z + \bar{m} + {\bar{n}}^{2}) + (\bar{m} + {\bar{n}}^{2})}{- \bar{n} z + \bar{m} + {\bar{n}}^{2}} = \frac{2 \bar{n}}{\bar{n} - k z} \end{matrix}

for any

z \in D

. Thus, the coefficient of

z^{2}

must be 0. This implies that

n = 0

, which is a contradiction. The proof is complete. □

In the remainder of this paper, we consider 2-complex symmetric composition operators with J which are induced linear fractional self-maps with

a \neq 0

c \neq 0

and

ϕ (0) \neq 0

and obtain that all composition operators are not 2-complex symmetric with J.

Theorem 6.

Let

ϕ (z) = \frac{a z + b}{c z + d}

be a linear fractional self-map of

D

such that

a \neq 0

c \neq 0

and

ϕ (0) \neq 0

. Then,

C_{ϕ}

is not 2-complex symmetric with J.

Proof.

We prove it by contradiction. Assume that

C_{ϕ}

is 2-complex symmetric with J. Since

a \neq 0

c \neq 0

and

ϕ (0) \neq 0

, set

ϕ (z) = \frac{m z + n}{s z + 1}

, where

m = \frac{a}{d}

n = \frac{b}{d}

and

s = \frac{c}{d}

. Then,

σ (z) = \frac{\bar{m} z - \bar{s}}{- \bar{n} z + 1}

g (z) = \frac{1}{- \bar{n} z + 1}

and

h (z) = s z + 1

. After a calculation, we obtain

ϕ_{2} (z) = \frac{(m^{2} + n s) z + m n + n}{(m s + s) z + n s + 1}, h (\frac{1}{\bar{σ (z)}}) = \frac{(m - n s) \bar{z}}{m \bar{z} - s}

and

ϕ (\bar{σ (z)}) = \frac{(m^{2} - n^{2}) \bar{z} + n - m s}{(m s - n) \bar{z} + 1 - s^{2}} .

Lemma 2 gives that

\begin{matrix} J C_{ϕ}^{2} K_{w} (z) & = K_{\bar{w}} (\bar{ϕ_{2} (\bar{z})}) = \frac{1}{1 - w \frac{({\bar{m}}^{2} + \bar{n s}) z + \bar{m n} + \bar{n}}{(\bar{m s} + \bar{s}) z + \bar{n s} + 1}} \\ = \frac{(\bar{m s} + \bar{s}) z + \bar{n s} + 1}{(\bar{m s} + \bar{s}) z + \bar{n s} + 1 - [({\bar{m}}^{2} + \bar{n s}) z + \bar{m n} + \bar{n}] w}, \end{matrix}

(22)

\begin{matrix} C_{ϕ}^{*} J C_{ϕ} K_{w} (z) = & - \bar{s} \frac{g (w)}{σ (w)} K_{ϕ (0)} (z) + \bar{h (\frac{1}{\bar{σ (w)}})} g (w) K_{ϕ (\bar{σ (w)})} (z) \\ = & \frac{- \bar{s}}{\bar{m} w - \bar{s}} \frac{1}{1 - \bar{n} z} + \frac{(\bar{m} - \bar{n s}) w}{\bar{m} w - \bar{s}} \frac{1}{- \bar{n} w + 1} \cdot \frac{1}{1 - \frac{({\bar{m}}^{2} - {\bar{n}}^{2}) w - \bar{m s} + \bar{n}}{(\bar{m s} - \bar{n}) w + 1 - {\bar{s}}^{2}} z} \\ = & \frac{- \bar{s}}{(\bar{m} w - \bar{s}) (1 - \bar{n} z)} + \frac{(\bar{m} - \bar{n s}) w}{(\bar{m} w - \bar{s}) (- \bar{n} w + 1)} \\ \cdot \frac{(\bar{m s} - \bar{n}) w + 1 - {\bar{s}}^{2}}{(\bar{m s} - \bar{n}) w + 1 - {\bar{s}}^{2} - [({\bar{m}}^{2} - {\bar{n}}^{2}) w - \bar{m s} + \bar{n}] z} \end{matrix}

(23)

and

\begin{matrix} {C_{ϕ}^{*}}^{2} J K_{w} (z) & = K_{ϕ_{2} (\bar{w})} (z) = \frac{1}{1 - z \frac{({\bar{m}}^{2} + \bar{n s}) w + \bar{m n} + \bar{n}}{(\bar{m s} + \bar{s}) w + \bar{n s} + 1}} \\ = \frac{(\bar{m s} + \bar{s}) w + \bar{n s} + 1}{(\bar{m s} + \bar{s}) w + \bar{n s} + 1 - [({\bar{m}}^{2} + \bar{n s}) w + \bar{m n} + \bar{n}] z} \end{matrix}

(24)

for any

w, z \in D

. Taking

w = 0

, then we obtain from (22)–(24) that

J C_{ϕ}^{2} K_{0} (z) = 1, C_{ϕ}^{*} J C_{ϕ} K_{0} (z) = \frac{1}{1 - \bar{n} z}

and

{C_{ϕ}^{*}}^{2} J K_{0} (z) = \frac{\bar{n s} + 1}{\bar{n s} + 1 - (\bar{m n} + \bar{n}) z}

for any

z \in D

. Since

C_{ϕ}

is 2-complex symmetric with J, we obtain that

\begin{matrix} 1 + \frac{\bar{n s} + 1}{\bar{n s} + 1 - (\bar{m n} + \bar{n}) z} = \frac{2}{1 - \bar{n} z} \end{matrix}

for any

z \in D

, which implies that

\begin{matrix} \frac{2 \bar{n s} + 2 - (\bar{m n} + \bar{n}) z}{\bar{n s} + 1 - (\bar{m n} + \bar{n}) z} = \frac{2}{1 - \bar{n} z} \end{matrix}

for any

z \in D

. Noting that the coefficient of

z^{2}

must be 0, then we have that

{\bar{n}}^{2} (m + 1) = 0

. Since

ϕ (0) \neq 0

, then

m = - 1

Noting that

ϕ_{2} (z) = \frac{(m^{2} + n s) z + m n + n}{(m s + s) z + n s + 1}

, then Lemma 4 gives that

2 ϕ_{2} (0) - 4 ϕ (0) + ϕ (0) ϕ_{2} (0) = \frac{2 (m n + n)}{n s + 1} - 4 n + \frac{n (m n + n)}{n s + 1} = 0 .

By a simple calculation, we see that

2 m - 4 n s + m n + n = 2

. Since

m = - 1

, we obtain that

n s = - 1

. Therefore,

m - n s = 0

, which means that

ϕ

is not a linear fractional self-map of

D

, a contradiction. The proof is complete. □

5. Conclusions

In this paper, we obtained some characterizations for composition operators

C_{ϕ}

to be 2-complex symmetric with respect to the conjugation J on the Hardy space

H^{2}

. To be specific, in Theorem 1, we obtain the necessary and sufficient condition for

C_{ϕ}

to be 2-complex symmetric with J when

ϕ

is an automorphism of

D

. Next, we discuss 2-complex symmetric composition operators

C_{ϕ}

which were induced by linear fractional self-maps of

D

with four different forms. The necessary and sufficient conditions for

C_{ϕ}

to be 2-complex symmetric with J in the first two cases are given in Theorems 2 and 3, respectively. In Theorems 5 and 6, we obtain that there is no 2-complex symmetric composition operators with J in the latter two cases.

Author Contributions

Conceptualization, S.L.; investigation, S.L., L.H. and R.Y.; writing—original draft preparation, L.H. and R.Y.; writing—review and editing, S.L. and L.H.; project administration, S.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflict of interest.

References

Garcia, S.; Putinar, M. Complex symmetric operators and applications. Trans. Am. Math. Soc. 2006, 358, 1285–1315. [Google Scholar] [CrossRef]
Garcia, S.; Putinar, M. Complex symmetric operators and applications, II. Trans. Am. Math. Soc. 2007, 359, 3913–3931. [Google Scholar] [CrossRef]
Garcia, S.; Wogen, W. Complex symmetric partial isometries. J. Funct. Anal. 2009, 257, 1251–1260. [Google Scholar] [CrossRef]
Garcia, S.; Wogen, W. Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 2010, 362, 6065–6077. [Google Scholar] [CrossRef]
Fatehi, M. Complex symmetric weighted composition operators. Complex Var. Elliptic Equ. 2019, 64, 710–720. [Google Scholar] [CrossRef]
Gupta, A.; Malhotra, A. Complex symmetric weighted composition operators on the space $H_{1}^{2}$ (D). Complex Var. Elliptic Equ. 2020, 65, 1488–1500. [Google Scholar] [CrossRef]
Gao, Y.; Zhou, Z. Complex symmetric composition operators induced by linear fractional maps. Indiana Univ. Math. J. 2020, 69, 367–384. [Google Scholar] [CrossRef]
Han, K.; Wang, M. Weighted composition-differentiation operators on the Hardy space. Banach J. Math. Anal. 2021, 15, 44. [Google Scholar] [CrossRef]
Jung, S.; Kim, Y.; Ko, E.; Lee, J. Complex symmetric weighted composition operators on H²(D). J. Funct. Anal. 2014, 267, 323–351. [Google Scholar] [CrossRef]
Lim, R.; Khoi, L. Complex symmetric weighted composition operators on H_γ(D). J. Math. Anal. Appl. 2018, 464, 101–118. [Google Scholar] [CrossRef]
Narayan, S.; Sievewright, D.; Thompson, D. Complex symmetric composition operators on H². J. Math. Anal. Appl. 2016, 443, 625–630. [Google Scholar] [CrossRef]
Narayan, S.; Sievewright, D.; Tjani, M. Complex symmetric composition operators on weighted Hardy spaces. Proc. Am. Math. Soc. 2020, 148, 2117–2127. [Google Scholar] [CrossRef]
Thompson, D.; McClatchey, T.; Holleman, C. Binormal, complex symmetric operators. Linear Mult. Algebra 2021, 69, 1705–1715. [Google Scholar] [CrossRef]
Yao, X. Complex symmetric composition operators on a Hilbert space of Dirichlet series. J. Math. Anal. Appl. 2017, 452, 1413–1419. [Google Scholar] [CrossRef]
Chō, M.; Ko, E.; Lee, J. On m-complex symmetric operators. Mediterr. J. Math. 2016, 13, 2025–2038. [Google Scholar] [CrossRef]
Exner, G.; Jin, J.; Jung, I.; Lee, J. On m-complex symmetric weighted shift operators on Cⁿ. Linear Algebra Appl. 2020, 603, 130–153. [Google Scholar] [CrossRef]
Chō, M.; Ko, E.; Lee, J. On m-complex symmetric operators, II. Mediterr. J. Math. 2016, 13, 3255–3264. [Google Scholar] [CrossRef]
Jeridi, N.; Rabaoui, R. On (A,m)-symmetric operators in a Hilbert space. Results Math. 2019, 74, 124. [Google Scholar] [CrossRef]
Cowen, C. Composition operators on H². J. Oper. Theory 1983, 9, 77–106. [Google Scholar]
Cowen, C.; MacCluer, B. Composition Operators on Spaces of Analytic Functions; Studies in Advanced Mathematics; CRC Press: Boca Raton, FL, USA, 1995; 388p, ISBN 0-8493-8492-3. [Google Scholar]
Garcia, S.; Hammond, C. which weighted composition operators are complex symmetric? Oper. Theory Adv. Appl. 2014, 236, 171–179. [Google Scholar]
Jung, S.; Kim, Y.; Ko, E. Composition operators for which $C_{ϕ}^{*}$ C_ϕ and C_ϕ + $C_{ϕ}^{*}$ commute. Complex Var. Elliptic Equ. 2014, 59, 1608–1625. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Hu, L.; Li, S.; Yang, R. 2-Complex Symmetric Composition Operators on H². Axioms 2022, 11, 358. https://doi.org/10.3390/axioms11080358

AMA Style

Hu L, Li S, Yang R. 2-Complex Symmetric Composition Operators on H². Axioms. 2022; 11(8):358. https://doi.org/10.3390/axioms11080358

Chicago/Turabian Style

Hu, Lian, Songxiao Li, and Rong Yang. 2022. "2-Complex Symmetric Composition Operators on H²" Axioms 11, no. 8: 358. https://doi.org/10.3390/axioms11080358

APA Style

Hu, L., Li, S., & Yang, R. (2022). 2-Complex Symmetric Composition Operators on H². Axioms, 11(8), 358. https://doi.org/10.3390/axioms11080358

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

2-Complex Symmetric Composition Operators on H²

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Linear Fractional Self-Maps

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Article Menu

2-Complex Symmetric Composition Operators on H2

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Linear Fractional Self-Maps

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

2-Complex Symmetric Composition Operators on H²