1 Introduction
Function composition is an operation that takes two functions f and
$\varphi $
and produces a function g such that
$g(x)=f(\varphi (x))$
. If f varies in a linear space of functions defined on the range of
$\varphi $
, then the mapping
$C_\varphi $
sending f into
$f\circ \varphi $
is a linear transformation, called a composition operator. Although the composition operation is basic to mathematics and its studies have been pursued for a long time, the study of composition operators as a part of operator theory has a relatively short history, starting in the mid 1960s with Nordgren’s paper [Reference Hai and Khoi16].
The books by Cowen and MacCluer [Reference Cowen and MacCluer4] and Shapiro [Reference Shapiro18] describe composition operators on Hardy and Bergman spaces. On Fock space, Carswell et al. [Reference Carswell, MacCluer and Schuster2] found that only the class of affine transformations
$\varphi (z)=az+b,|a|\leq 1$
and
$b=0$
whenever
$|a|=1$
induces bounded composition operators. They characterised compactness of
$C_\varphi $
by the strict requirement
$|a|<1$
. In [Reference Dieu and Ohno14], Le described all bounded composition operators that are normal. Later, the author [Reference Hosokawa, Nieminen and Ohno10] extended Le’s result to unbounded composition operators and also obtained conditions that are more general than the bounded case.
Berkson [Reference Berkson1] initiated the study of a sum of composition operators and this was taken further by Shapiro and Sundberg [Reference Shapiro and Sundberg19], who investigated the topological structure of the space of composition operators acting on Hardy space. For Fock spaces, Choe et al. [Reference Choe, Izuchi and Koo3] showed that a linear sum of two composition operators is bounded (respectively, compact) if and only if both composition operators are already bounded (respectively, compact). Other properties have been studied such as conditions for a sum to be bounded on Bloch spaces [Reference Le12] or completely continuous on Hardy and Bergman spaces [Reference Garcia, Hammond, Boiso, Hedenmalm, Kaashock, Rodríguez and Treil5].
A second source of our study is the theory of complex symmetric operators initiated by Garcia and Putinar [Reference Garcia and Putinar8, Reference Hai9]. A complex symmetric operator is an unbounded, linear operator
$T:\mathrm {dom}(T)\subseteq \mathcal H\to \mathcal H$
on a Hilbert space
$\mathcal H$
with the property that
$T=\mathcal C T^*\mathcal C$
, where
$\mathcal C$
is an isometric involution (in short, conjugation) on
$\mathcal H$
. To indicate the dependence on conjugation, this case is called
$\mathcal C$
-selfadjoint. A long list of well-known operators have been proven complex symmetric: normal operators, Hankel matrices and compressed Toeplitz operators (including the compressed shift). For more on complex symmetry, see [Reference Garcia, Prodan and Putinar7–Reference Garcia and Putinar9].
Recently, there has been interest in the problem of classifying composition operators that are complex symmetric. The papers [Reference Garcia, Prodan and Putinar6, Reference Mead13] studied the problem on Hardy spaces of the unit disk corresponding to the conjugation
$$ \begin{gather*} \mathcal Q f(z)=\overline{f(\overline{z})}. \end{gather*} $$
The conjugation
$\mathcal Q$
inspired the author and Khoi [Reference Jung, Kim, Ko and Lee11] to study this problem in Fock space.
In this paper, we are interested in unbounded, linear operators arising from the expression
$\mathcal E(\varphi ,d)f=\sum _{j=1}^d f\circ \varphi _j$
, where the functions satisfy
$$ \begin{align} \varphi_t\not\equiv\varphi_s \quad \text{whenever}\ t\ne s. \end{align} $$
Our research is conducted on Fock space
$$ \begin{gather*} \mathcal F^2=\bigg\{f(z)=\sum_{n=0}^{\infty}f_nz^n\,\,\text{is entire with}\,\,\sum_{n=0}^{\infty}n!|f_n|^2<\infty\bigg\}, \end{gather*} $$
the reproducing kernel Hilbert space with the reproducing kernel
$K_z^{[m]}(u)=u^me^{u\overline {z}}$
, equipped with the inner product
$$ \begin{gather*} \langle f,g\rangle=\sum_{n=0}^{\infty}n!f_n\overline{g_n},\quad\text{where}\ f(z)=\sum_{n=0}^{\infty}f_nz^n,\ g(z)=\sum_{n=0}^{\infty}g_nz^n. \end{gather*} $$
The maximal operator corresponding to
$\mathcal E(\varphi ,d)$
over
$\mathcal F^2$
is defined by
$$ \begin{gather*} \mathrm{dom}(\mathcal S_{\varphi,d,\max})=\{f\in\mathcal F^2:\mathcal E(\varphi,d)f\in\mathcal F^2\}, \\ \mathcal S_{\varphi,d,\max}f=\mathcal E(\varphi,d)f\quad\mbox{for all } f\in\mathrm{dom}(\mathcal S_{\varphi,d,\max}). \end{gather*} $$
The operator
$\mathcal S_{\varphi ,d}$
is called a nonmaximal operator if
$\mathcal S_{\varphi ,d}\preceq \mathcal S_{\varphi ,d,\max }$
. We characterise the real symmetry and complex symmetry of
$\mathcal S_{\varphi ,d}$
.
2 Some initial properties
We list notation used in the present paper. Let
$[1,d]_{\mathbb Z}=\{\kern1.2pt j\in \mathbb Z:1\leq j\leq d\}$
. The symbol
${\mathbb C}_k[z]$
denotes the set of all polynomials with degree at most k and complex coefficients. For two unbounded operators
$A,B$
, writing
$A\preceq B$
means that
$\mathrm {dom}(A)\subseteq \mathrm {dom}(B)$
and
$Ax=Bx$
for
$x\in \mathrm {dom}(A)$
.
2.1 Elementary observations
These observations are used in Lemmas 3.1 and 4.1.
Lemma 2.1. Let
$\{C_j:j\in [1,d]_{\mathbb Z}\}$
and
$\{D_j:j\in [1,d]_{\mathbb Z}\}$
be sets of distinct numbers. Let
$\lambda _1,\ldots ,\lambda _d,\gamma _1,\ldots ,\gamma _d$
be nonzero complex numbers. If the equality
$$ \begin{gather*} \sum\limits_{j=1}^d\lambda_j K_{C_j} =\sum\limits_{j=1}^d\gamma_j K_{D_j} \end{gather*} $$
holds, then for each
$t\in [1,d]_{\mathbb Z}$
there exists a unique
$s\in [1,d]_{\mathbb Z}$
such that
$$ \begin{gather*} C_t=D_s,\quad\lambda_t=\gamma_s. \end{gather*} $$
Proof
Assume, towards a contradiction, that there is
$t\in [1,d]_{\mathbb Z}$
for which
$C_t\ne D_j$
for every
$j\in [1,d]_{\mathbb Z}$
. Set
$$ \begin{gather*} f(z)=\prod_{\ell=1}^d (z-D_\ell)\cdot\prod_{\mu\ne t}(z-C_\mu). \end{gather*} $$
It is clear that
$f\in \mathcal F^2$
and
$f(C_t)\ne 0$
. We have
$$ \begin{gather*} \overline{\lambda_t}f(C_t)=\bigg\langle {f}, {\sum\limits_{j=1}^d\lambda_j K_{C_j}} \bigg\rangle =\bigg\langle {f}, {\sum\limits_{j=1}^d\gamma_j K_{D_j}} \bigg\rangle=0; \end{gather*} $$
but this is impossible. Thus, there exists
$s\in [1,d]_{\mathbb Z}$
such that
$C_t=D_s$
. Since the set
$\{D_t:t\in [1,d]_{\mathbb Z}\}$
consists of distinct numbers, this number s is unique. Setting
$$ \begin{gather*} g(z)=\prod_{\ell\ne s}(z-D_\ell)\cdot\prod_{\mu\ne t}(z-C_\mu), \end{gather*} $$
$$ \begin{gather*} \overline{\lambda_t} g(C_t)=\bigg\langle {g}, {\sum\limits_{j=1}^d\lambda_j K_{C_j}} \bigg\rangle =\bigg\langle {g}, {\sum\limits_{j=1}^d\gamma_j K_{D_j}} \bigg\rangle=\overline{\gamma_s}g(D_s)=\overline{\gamma_s}g(C_t), \end{gather*} $$
which implies, as
$g(C_t)\ne 0$
, that
$\lambda _t=\gamma _s$
.
2.2 Action on kernel functions
The next lemma describes how the adjoint
$\mathcal S_{\varphi ,d}^*$
acts on kernel functions.
Lemma 2.2. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions; that is, condition (1.1) holds. Let
$\mathcal S_{\varphi ,d}$
be the densely defined operator arising from the expression
$\mathcal E(\varphi ,d)$
. Then for every
$z\in {\mathbb C}$
,
$K_z\in \mathrm {dom}(\mathcal S_{\varphi ,d}^*)$
and
$$ \begin{align*} \mathcal S_{\varphi,d}^*K_z=\sum_{j=1}^dK_{\varphi_j(z)}. \end{align*} $$
Proof
For every
$f\in \mathrm {dom}(\mathcal S_{\varphi ,d})$
,
$$ \begin{align*} \langle \mathcal S_{\varphi,d}f,K_z\rangle &= \mathcal S_{\varphi,d}f(z)=\sum_{j=1}^d\langle f,K_{\varphi_j(z)}\rangle=\bigg\langle f,\sum_{j=1}^dK_{\varphi_j(z)}\bigg\rangle, \end{align*} $$
which gives the desired conclusion.
2.3 Closed operators
As it turns out, the maximal operator is always closed.
Proposition 2.3. The maximal operator
$\mathcal S_{\varphi ,d,\max }$
is always closed on Fock space
$\mathcal F^2$
.
Proof
Let
$(u_n)\subset \mathcal F^2$
and
$u,v\in \mathcal F^2$
be such that
$u_n\to u \text{ and }\mathcal S_{\varphi,d,\max}u_n\to v \mbox{ in } \mathcal F^2$
. It follows that
$u_n(z)\to u(z) \text{ and } \mathcal S_{\varphi,d,\max}u_n(z)\to v(z) \mbox{ for all } z\in{\mathbb C}$
and, consequently,
$$ \begin{align*}\mathcal S_{\varphi,d,\max}u_n(z)=\sum_{j=1}^d u_n(\varphi_j(z))\to\sum_{j=1}^d u(\varphi_j(z))\quad\mbox{for all } z\in{\mathbb C}. \end{align*} $$
Thus,
$$ \begin{align*}\sum_{j=1}^d u(\varphi_j(z))=v(z)\quad\mbox{for all } z\in {\mathbb C}. \end{align*} $$
Since
$v\in \mathcal F^2$
, we conclude that
$u\in \mathrm {dom}(\mathcal S_{\varphi ,d,\max })$
and
$\mathcal S_{\varphi ,d,\max }u=v$
.
Corollary 2.4. The maximal operator
$\mathcal S_{\varphi ,d,\max }$
is bounded on Fock space
$\mathcal F^2$
if and only if its domain
$\mathrm {dom}(\mathcal S_{\varphi ,d,\max })=\mathcal F^2$
.
2.4 Dense domain
In the next result, we characterise the maximal operator
$\mathcal S_{\varphi ,d,\max }$
when it is densely defined.
Proposition 2.5. Let
$\mathcal R$
be the linear operator defined by
$$ \begin{align*}\mathrm{dom}(\mathcal R)=\mathrm{span}\{K_x:x\in{\mathbb C}\},\quad\mathcal R K_x=\sum_{j=1}^d K_{\varphi_j(x)}. \end{align*} $$
Then
$\mathcal S_{\varphi ,d,\max }=\mathcal R^*$
. Moreover, the operator
$\mathcal S_{\varphi ,d,\max }$
is densely defined if and only if the operator
$\mathcal R$
is closable.
Proof
Let
$u=\sum _{j=1}^n\lambda _j K_{x_j}\in \mathrm {dom}(\mathcal R)$
. For every
$v\in \mathcal F^2$
,
$$ \begin{gather*} \langle{\mathcal R u},{v}\rangle =\sum_{j=1}^n\sum_{t=1}^d\lambda_j\langle{K_{\varphi_t(x_j)}},{v}\rangle =\sum_{j=1}^n\sum_{t=1}^d\lambda_j\overline{v(\varphi_t(x_j))} =\sum_{j=1}^n\lambda_j\overline{\mathcal E(\varphi,d)v(x_j)}. \end{gather*} $$
By the Riesz lemma,
$v\in \mathrm {dom}(\mathcal R^*)$
if and only if there exists
$\omega =\omega (v)>0$
such that
$$ \begin{gather*} |\langle \mathcal R u,v\rangle|\leq\omega\|u\|\quad\mbox{for all } u\in\mathrm{dom}(\mathcal R) \end{gather*} $$
or, equivalently,
$$ \begin{gather*} \bigg|\sum_{j=1}^n\mathcal E(\varphi,d)v(x_j)\overline{\lambda_j}\bigg|^2\leq\omega^2\sum_{j,\ell=1}^n \lambda_j\overline{\lambda_\ell}K_{x_j}(x_\ell). \end{gather*} $$
It was proven in [Reference Szafraniec and Alpay20] that this is equivalent to saying that
$\mathcal E(\varphi ,d)v\in \mathcal F^2$
. Consequently, the domain
$\mathrm {dom}(\mathcal R^*)=\mathrm {dom}(\mathcal S_{\varphi ,d,\max })$
and
$$ \begin{gather*} \langle \mathcal R u,v\rangle=\langle u,\mathcal E(\varphi,d)v\rangle=\langle u,\mathcal S_{\varphi,d,\max}v\rangle \quad\mbox{for all } u\in\mathrm{dom}(\mathcal R),\ v\in\mathrm{dom}(\mathcal S_{\varphi,d,\max}), \end{gather*} $$
which yields
$\mathcal S_{\varphi ,d,\max }=\mathcal R^*$
. The second assertion follows from [Reference Schmüdgen17, Proposition 1.8(i)].
3 Real symmetry
An unbounded, linear operator T is called real symmetric on a complex, separable Hilbert space if the equality
$T=T^*$
holds; meaning that
$\mathrm {dom}(T)=\mathrm {dom}(T^*)$
and
$Tx=T^*x$
for
$x\in \mathrm {dom}(T)$
.
In this section, we are interested in how the real symmetry of
$\mathcal S_{\varphi ,d}$
impacts the function-theoretic properties of
$\varphi _j$
, for
$j\in [1,d]_{\mathbb Z}$
, and vice versa. For the necessary condition, we apply the real symmetry to kernel functions and this step leads to Lemma 3.1. As it turns out, the real symmetry significantly restricts the possible functions for the operator
$\mathcal S_{\varphi ,d}$
. For the sufficient condition, we need a computation regarding the adjoint
$\mathcal S_{\varphi ,d}^*$
.
Lemma 3.1. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions, that is, condition (1.1) holds. Suppose that
$$ \begin{align} \sum\limits_{j=1}^d e^{x\overline{\varphi_j(z)}} =\sum\limits_{j=1}^d e^{\varphi_j(x)\overline{z}}\quad\mbox{for all } x,z\in{\mathbb C}. \end{align} $$
Then the functions
$\varphi _j$
have the form
$$ \begin{align} \varphi_j(z)=A_jz,\quad j\in[1,d]_{\mathbb Z}, \end{align} $$
where the coefficients satisfy the two conditions
$$ \begin{align} \{A_j:j\in [1,d]_{\mathbb Z}\}=\{\overline{A_j}:j\in [1,d]_{\mathbb Z}\} \end{align} $$
and
$$ \begin{align} A_t\ne A_s\quad\mbox{for all } t\ne s. \end{align} $$
Proof
Step 1: We show that for each
$k\in [1,d]_{\mathbb Z}$
,
$$ \begin{align} \sum\limits_{j=1}^d\overline{\varphi_j(z)}^k e^{x\overline{\varphi_j(z)}} =\sum\limits_{j=1}^d\bigg(\sum\limits_{t=1}^k\omega_{t,j,k}(x)\overline{z}^t\bigg)\,e^{\varphi_j(x)\overline{z}}\quad\mbox{for all } x,z\in{\mathbb C}, \end{align} $$
where
$$ \begin{align*} \begin{array}{rll} \omega_{1,j,1}(x)\!\!\!\!&=\varphi_j'(x)&\ \text{if}\ \ j\in[1,d]_{\mathbb Z},\\ \omega_{1,j,k+1}(x)\!\!\!\!&=\omega_{1,j,k}'(x)&\ \text{if}\ \ j\in[1,d]_{\mathbb Z},\\ \omega_{t,j,k+1}(x)\!\!\!\!&=\omega_{t,j,k}'(x)+\varphi_j'(x)\omega_{t-1,j,k}(x)&\ \text{if}\ \ t\in[2,k]_{\mathbb Z} \ \text{ and }\ j\in[1,d]_{\mathbb Z},\\ \omega_{k+1,j,k+1}(x)\!\!\!\!&=\varphi_j'(x)\omega_{k,j,k}(x)&\ \text{if}\ \ j\in[1,d]_{\mathbb Z}. \end{array} \end{align*} $$
Indeed, differentiating (3.1) with respect to the variable x,
$$ \begin{gather*} \sum\limits_{j=1}^d\overline{\varphi_j(z)} e^{x\overline{\varphi_j(z)}} =\sum\limits_{j=1}^d \varphi_j'(x)\overline{z}e^{\varphi_j(x)\overline{z}} =\sum\limits_{j=1}^d \omega_{1,j,1}(x)\overline{z}e^{\varphi_j(x)\overline{z}}, \end{gather*} $$
which means that equality (3.5) holds when
$k=1$
. Now suppose that (3.5) holds for k and consider
$k+1$
. We have
$$ \begin{gather*} \sum\limits_{j=1}^d\overline{\varphi_j(z)}^k e^{x\overline{\varphi_j(z)}} =\sum\limits_{j=1}^d\bigg(\sum\limits_{t=1}^k\omega_{t,j,k}(x)\overline{z}^t\bigg)\,e^{\varphi_j(x)\overline{z}}, \end{gather*} $$
which implies, after differentiating with respect to the variable x, that
$$ \begin{align*} \sum\limits_{j=1}^d\overline{\varphi_j(z)}^{k+1} e^{x\overline{\varphi_j(z)}} &=\sum\limits_{j=1}^d \bigg(\sum\limits_{t=1}^k\omega_{t,j,k}'(x)\overline{z}^t +\varphi_j'(x)\overline{z}\sum\limits_{t=1}^k\omega_{t,j,k}(x)\overline{z}^t\bigg) e^{\varphi_j(x)\overline{z}}\\ &=\sum\limits_{j=1}^d\bigg(\sum\limits_{t=1}^{k+1}\omega_{t,j,k+1}(x)\overline{z}^t\bigg)\,e^{\varphi_j(x)\overline{z}}. \end{align*} $$
Step 2: We claim that
$\varphi _j(0)=0$
for every
$j\in [1,d]_{\mathbb Z}$
. Assume, towards a contradiction, that
$\varphi _t(0)\ne 0$
for some
$t\in [1,d]_{\mathbb Z}$
. Set
$$ \begin{gather*} \Omega_t=\{\kern1.2pt j:\varphi_j(0)=\varphi_t(0)\},\quad f(z)=z\prod_{j\notin\Omega_t}(z-\varphi_j(0)). \end{gather*} $$
Letting
$x=0$
in (3.1), we find that
$\sum _{j=1}^d K_{\varphi _j(0)}=d=dK_0$
and so
$$ \begin{gather*} 0=d\langle{f},{K_0}\rangle=\sum_{j=1}^d\langle{f},{K_{\varphi_j(0)}}\rangle =\bigg(\sum_{j\in\Omega_t}+\sum_{j\notin\Omega_t}\bigg)f(\varphi_j(0))=|\Omega_t|f(\varphi_t(0)), \end{gather*} $$
where the notation
$|\Omega _t|$
stands for the cardinality of
$\Omega _t$
; but this is impossible.
Step 3: We claim that
$$ \begin{align} \sum_{1\leq j_1<j_2<\cdots<j_k\leq d}\,\,\, \prod_{\ell=1}^k\varphi_{j_\ell}(z)=(-1)^kq_{d-k}(z)\quad\text{for some}\ q_m\in{\mathbb C}_{d-m}[z]. \end{align} $$
Let
$k\in [1,d]_{\mathbb Z}$
. Evaluating (3.5) at
$x=0$
and using
$\varphi _j(0)=0$
from Step 2 gives
$\sum _{j=1}^d\varphi _j(z)^k=p_k(z)$
for some
$p_k\in {\mathbb C}_k[z]$
. Denote the left-hand side of (3.6) by
$\widetilde {q}_k(z)$
. By Newton’s identities [Reference Nordgren15],
$$ \begin{gather*} k\widetilde{q}_k(z)=\sum_{j=1}^{k-1}(-1)^{j-1}\widetilde{q}_{k-j}(z)p_j(z). \end{gather*} $$
By induction on k, we have
$\widetilde {q}_k\in {\mathbb C}_k[z]$
. Setting
$q_{d-k}(z)=(-1)^k\widetilde {q}_k(z)$
gives (3.6).
With these preparations in place, we prove the conclusion of the lemma as follows. By Vieta’s formulas,
$\varphi _1(z),\ldots ,\varphi _d(z)$
are solutions of the algebraic equation
$$ \begin{gather*} X^d+\sum_{j=0}^{d-1}q_j(z)X^j=0. \end{gather*} $$
Hence, for
$|z|>R$
sufficiently large, we can find a constant
$C=C(R)$
with the property that
$$ \begin{gather*} |X|^d\leq C|z|^d\bigg(\sum_{j=0}^{d-1}|X|^j\bigg)\leq C|z|^d(1+|X|)^{d-1}. \end{gather*} $$
By Liouville’s theorem, this means that the functions
$\varphi _1,\ldots ,\varphi _d$
are polynomials with degrees at most d. Since
$\varphi _t(0)=0$
, we can write
$\varphi _t(z)=zg_t(z)$
, where
$g_t\in {\mathbb C}_{d-1}[z]$
. Since the product
$$ \begin{gather*} \varphi_1(z)\cdots\varphi_d(z)=z^d\prod_{j=1}^d g_j(z) \end{gather*} $$
lies in
${\mathbb C}_d[z]$
, the polynomials
$g_j$
are constants, giving the required form for the
$\varphi _j$
in (3.2). Substituting these forms back into (3.1) gives
$$ \begin{gather*} \sum\limits_{j=1}^d e^{x\overline{A_jz}} =\sum\limits_{j=1}^d e^{A_jx\overline{z}}\quad\mbox{for all } x,z\in{\mathbb C}. \end{gather*} $$
Consequently, taking into account the explicit form of the kernel functions,
$$ \begin{gather*} \sum\limits_{j=1}^d K_{A_j}(u) =\sum\limits_{j=1}^d K_{\overline{A_j}}(u)\quad\mbox{for all } u\in{\mathbb C}. \end{gather*} $$
Now we can apply Lemma 2.1 to get (3.3). Finally, (3.4) follows from (1.1).
The next result concerns real symmetric operators with maximal domains.
Theorem 3.2. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions, that is, condition (1.1) holds. Let
$\mathcal S_{\varphi ,d,\max }$
be the maximal operator arising from the expression
$\mathcal E(\varphi ,d)$
. Then the operator
$\mathcal S_{\varphi ,d,\max }$
is real symmetric if and only if the functions
$\varphi _j$
have the form (3.2) with conditions (3.3) and (3.4).
Proof
Suppose that the operator
$\mathcal S_{\varphi ,d,\max }$
is real symmetric. This implies, in particular, that
$$ \begin{gather*} \mathcal S_{\varphi,d,\max}^*K_z(x)=\mathcal S_{\varphi,d,\max}K_z(x)\quad\mbox{for all } x,z\in{\mathbb C} \end{gather*} $$
and, by Lemma 3.1, we get the necessary condition.
For the sufficient condition, take the functions as in the statement of the theorem. A computation shows that
$K_z\in \mathrm {dom}(\mathcal S_{\varphi ,d,\max })$
and moreover
$$ \begin{gather*} \mathcal S_{\varphi,d,\max}K_z(x)=\sum_{j=1}^dK_{\overline{A_j}z}(x)\quad\mbox{for all } z,x\in{\mathbb C}. \end{gather*} $$
First, we prove the inclusion
$$ \begin{align} \mathcal S_{\varphi,d,\max}^*\preceq\mathcal S_{\varphi,d,\max}. \end{align} $$
Indeed, for
$u\in \mathrm {dom}(\mathcal S_{\varphi ,d,\max }^*)$
and
$z\in {\mathbb C}$
,
$$ \begin{align*} \mathcal S_{\varphi,d,\max}^*u(z) &=\langle{\mathcal S_{\varphi,d,\max}^*u},{K_z}\rangle=\langle{u},{\mathcal S_{\varphi,d,\max}K_z}\rangle \\ &=\bigg\langle {u}, {\sum_{j=1}^dK_{\overline{A_j}z}} \bigg\rangle=\bigg\langle {u}, {\sum_{j=1}^dK_{A_jz}} \bigg\rangle\quad\text{(by (3.3))}\\ & =\sum_{j=1}^du(A_jz)=\mathcal S_{\varphi,d,\max}u(z). \end{align*} $$
Next, we show that equality occurs in (3.7), meaning that
$$ \begin{gather*} \langle{\mathcal S_{\varphi,d,\max}g},{h}\rangle=\langle{g},{\mathcal S_{\varphi,d,\max}h}\rangle \quad\mbox{for all } g,h\in\mathrm{dom}(\mathcal S_{\varphi,d,\max}). \end{gather*} $$
Take arbitrary
$g(z)=\sum _{n=0}^{\infty }g_nz^n, h(z)=\sum _{n=0}^{\infty }h_nz^n \in \mathrm {dom}(\mathcal S_{\varphi ,d,\max })$
with Taylor coefficients
$g_n,h_n\in {\mathbb C}$
. We have
$$ \begin{gather*} \mathcal S_{\varphi,d,\max}g(z) =\sum_{j=1}^d\sum_{n=0}^{\infty}g_nA_j^nz^n=\sum_{n=0}^{\infty}g_n\bigg(\sum_{j=1}^dA_j^n\bigg)z^n. \end{gather*} $$
Since
$(z^n)$
is an orthogonal basis,
$$ \begin{gather*} \langle{\mathcal S_{\varphi,d,\max}g},{h}\rangle =\sum_{n=0}^{\infty}g_n\overline{h_n}\bigg(\sum_{j=1}^dA_j^n\bigg)\|z^n\|^2 =\sum_{n=0}^{\infty}g_n\overline{h_n}\bigg(\sum_{j=1}^d\overline{A_j}^n\bigg)\|z^n\|^2 =\langle{g},{\mathcal S_{\varphi,d,\max}h}\rangle.\quad \\[-45pt] \end{gather*} $$
The next result relaxes the domain to show that the real symmetry cannot be separated from the maximality of the domain.
Theorem 3.3. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions, that is, condition (1.1) holds. Let
$\mathcal S_{\varphi ,d}$
be the operator arising from the expression
$\mathcal E(\varphi ,d)$
. Then the following assertions are equivalent.
-
(1) The operator
$\mathcal S_{\varphi ,d}$
is real symmetric. -
(2) The operator
$\mathcal S_{\varphi ,d}$
satisfies both the following conditions:
Proof
The implication (2)
$\Longrightarrow $
(1) follows directly from Theorem 3.2. We prove the reverse implication (1)
$\Longrightarrow $
(2) as follows. Suppose that the operator
$\mathcal S_{\varphi ,d}$
is real symmetric. Since
$\mathcal S_{\varphi ,d}\preceq \mathcal S_{\varphi ,d,\max }$
, it follows from [Reference Schmüdgen17, Proposition 1.6] that
$$ \begin{align*}\mathcal S_{\varphi,d,\max}^*\preceq \mathcal S_{\varphi,d}^*=\mathcal S_{\varphi,d}\preceq\mathcal S_{\varphi,d,\max}. \end{align*} $$
In particular,
$$ \begin{align*}\mathcal S_{\varphi,d,\max}^*K_z(x)=\mathcal S_{\varphi,d,\max}K_z(x)\quad\mbox{for all } z,x\in{\mathbb C}. \end{align*} $$
Lemma 3.1 yields (2b). Hence, by Theorem 3.2, the operator
$\mathcal S_{\varphi ,d,\max }$
is real symmetric. Then (2a) follows from
$$ \begin{align*} \mathcal S_{\varphi,d}\preceq \mathcal S_{\varphi,d,\max}=\mathcal S_{\varphi,d,\max}^*\preceq \mathcal S_{\varphi,d}^*=\mathcal S_{\varphi,d}.\\[-37pt] \end{align*} $$
4 Complex symmetry
In this section, we describe precisely when the functions
$\varphi _j,j\in [1,d]_{\mathbb Z}$
ensure that the operator
$\mathcal S_{\varphi ,d}$
is complex symmetric corresponding to the conjugation
$$ \begin{gather*} \mathcal Q_\omega f(z)=\overline{f(\omega\overline{z})}. \end{gather*} $$
As a consequence, we obtain the interesting fact that real symmetry implies complex symmetry; namely, if the operator
$\mathcal S_{\varphi ,d}$
is real symmetric, then it is complex symmetric with respect to
$\mathcal Q_\omega $
.
We start this section with an algebraic observation that is analogous to Lemma 3.1. We include a proof for the sake of completeness.
Lemma 4.1. Let
$\omega \in \mathbb T$
. Suppose that condition (1.1) holds. If
$$ \begin{align} \sum\limits_{j=1}^d e^{\overline{\omega}x\varphi_j(\omega\overline{z})} =\sum\limits_{j=1}^d e^{\varphi_j(x)\overline{z}}\quad\mbox{for all } x,z\in{\mathbb C}, \end{align} $$
then the functions
$\varphi _j$
have the form (3.2) with condition (3.4).
Proof
Letting
$x=0$
in (4.1),
$$ \begin{gather*} \sum\limits_{j=1}^d K_{\overline{\varphi_j(0)}}=d=dK_0. \end{gather*} $$
Using arguments similar to those used in Step 2 of Lemma 3.1, we also have
$\varphi _j(0)=0$
for
$j\in [1,d]_{\mathbb Z}$
.
For
$k\in [1,d]_{\mathbb Z}$
, differentiating (4.1) k times with respect to the variable x and then evaluating it at the point
$x=0$
gives
$\sum _{j=1}^d\varphi _j(z)^k=p_k(z)$
for some
$p_k\in {\mathbb C}_k[z]$
. By an inductive argument, we can find polynomials
$q_m\in {\mathbb C}_{d-m}[z]$
for which
$$ \begin{gather*} \sum_{1\leq j_1<j_2<\cdots<j_k\leq d}\,\,\, \prod_{\ell=1}^k\varphi_{j_\ell}(z)=(-1)^kq_{d-k}(z). \end{gather*} $$
By Vieta’s formulas,
$\varphi _1(z),\ldots ,\varphi _d(z)$
are solutions of the equation
$$ \begin{gather*} X^d+\sum_{j=0}^{d-1}q_j(z)X^j=0. \end{gather*} $$
Hence, for
$|z|>R$
large enough,
$|X|^d\leq C|z|^d(1+|X|)^{d-1}$
, which means that the functions
$\varphi _1,\ldots ,\varphi _d$
are polynomials with degrees at most d. Since
$\varphi _t(0)=0$
, we can write
$\varphi _t(z)=zg_t(z)$
, where
$g_t\in {\mathbb C}_{d-1}[z]$
. Since
$\varphi _1(z)\cdots \varphi _d(z)\in {\mathbb C}_d[z]$
, the polynomials
$g_j$
are constant. Hence, the functions
$\varphi _j$
have the form in (3.2). Condition (3.4) follows directly from (1.1).
Now we use Lemma 4.1 to characterise maximal operators that are complex symmetric corresponding to the conjugation
$\mathcal Q_\omega $
.
Theorem 4.2. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions, that is, condition (1.1) holds. Let
$\mathcal S_{\varphi ,d,\max }$
be the maximal operator arising from the expression
$\mathcal E(\varphi ,d)$
. Then the operator
$\mathcal S_{\varphi ,d,\max }$
is
$\mathcal Q_\omega $
-selfadjoint if and only if the functions are of the form in (3.2) with condition (3.4).
Proof
Suppose that the operator
$\mathcal S_{\varphi ,d,\max }$
is
$\mathcal Q_\omega $
-selfadjoint. This implies, in particular, that
$$ \begin{gather*} \mathcal Q_\omega\mathcal S_{\varphi,d,\max}^*\mathcal Q_\omega K_z(x)=\mathcal S_{\varphi,d,\max}K_z(x)\quad\mbox{for all } x,z\in{\mathbb C} \end{gather*} $$
and the necessary condition follows from Lemma 4.1.
For the sufficient condition, take functions as in the statement of the theorem. A computation shows that
$K_z\in \mathrm {dom}(\mathcal S_{\varphi ,d,\max })$
and moreover
$$ \begin{gather*} \mathcal Q_\omega\mathcal S_{\varphi,d,\max}\mathcal Q_\omega K_z(x)=\sum_{j=1}^dK_{A_jz}(x)\quad\mbox{for all } z,x\in{\mathbb C}. \end{gather*} $$
First, we prove that
$$ \begin{align} \mathcal Q_\omega\mathcal S_{\varphi,d,\max}^*\mathcal Q_\omega\preceq\mathcal S_{\varphi,d,\max}. \end{align} $$
Indeed, for
$u\in \mathrm {dom}(\mathcal S_{\varphi ,d,\max }^*)$
and
$z\in {\mathbb C}$
,
$$ \begin{align*} \mathcal Q_\omega\mathcal S_{\varphi,d,\max}^*\mathcal Q_\omega u(z) &=\langle{\mathcal Q_\omega\mathcal S_{\varphi,d,\max}^*\mathcal Q_\omega u},{K_z}\rangle =\langle{u},{\mathcal Q_\omega\mathcal S_{\varphi,d,\max}\mathcal Q_\omega K_z}\rangle \\ &=\bigg\langle {u}, {\sum_{j=1}^dK_{A_jz}} \bigg\rangle =\sum_{j=1}^du(A_jz)=\mathcal S_{\varphi,d,\max}u(z). \end{align*} $$
Next, we show that equality occurs in (4.2), meaning that
$$ \begin{gather*} \langle{\mathcal Q_\omega\mathcal S_{\varphi,d,\max}g},{h}\rangle=\langle{\mathcal Q_\omega\mathcal S_{\varphi,d,\max}h},{g}\rangle \quad \mbox{for all } g,h\in\mathrm{dom}(\mathcal S_{\varphi,d,\max}). \end{gather*} $$
Take arbitrary
$g(z)=\sum _{n=0}^{\infty }g_nz^n, h(z)=\sum _{n=0}^{\infty }h_nz^n \in \mathrm {dom}(\mathcal S_{\varphi ,d,\max })$
with Taylor coefficients
$g_n,h_n\in {\mathbb C}$
. We have
$$ \begin{gather*} \mathcal Q_\omega\mathcal S_{\varphi,d,\max}g(z) =\sum_{j=1}^d\sum_{n=0}^{\infty}\overline{g_nA_j^n\omega^n}z^n =\sum_{n=0}^{\infty}\overline{g_n\omega^n}\bigg(\sum_{j=1}^d\overline{A_j}^n\bigg)z^n. \end{gather*} $$
Since
$(z^n)$
is an orthogonal basis,
$$ \begin{align*} \langle{\mathcal Q_\omega\mathcal S_{\varphi,d,\max}g},{h}\rangle =\sum_{n=0}^{\infty}\overline{g_nh_n\omega^n}\bigg(\sum_{j=1}^d\overline{A_j}^n\bigg)\|z^n\|^2 =\langle{\mathcal Q_\omega\mathcal S_{\varphi,d,\max}h},{g}\rangle. \\[-45pt] \end{align*} $$
As with real symmetry, a complex symmetric operator must be maximal.
Theorem 4.3. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions, that is, condition (1.1) holds. Let
$\mathcal S_{\varphi ,d}$
be the operator arising from the expression
$\mathcal E(\varphi ,d)$
. Then the following assertions are equivalent.
-
(1) The operator
$\mathcal S_{\varphi ,d}$
is
$\mathcal Q_\omega $
-selfadjoint. -
(2) The operator
$\mathcal S_{\varphi ,d}$
satisfies both the following conditions:
Proof
The implication (2)
$\Longrightarrow $
(1) follows directly from Theorem 4.2. We prove the reverse implication (1)
$\Longrightarrow $
(2) as follows. Suppose that the operator
$\mathcal S_{\varphi ,d}$
is
$\mathcal Q_\omega $
-selfadjoint. Since
$\mathcal S_{\varphi ,d}\preceq \mathcal S_{\varphi ,d,\max }$
, it follows from [Reference Schmüdgen17, Proposition 1.6] that
$$ \begin{align*}\mathcal S_{\varphi,d,\max}^*\preceq \mathcal S_{\varphi,d}^*=\mathcal Q_\omega\mathcal S_{\varphi,d}\mathcal Q_\omega \preceq\mathcal Q_\omega\mathcal S_{\varphi,d,\max}\mathcal Q_\omega. \end{align*} $$
Thus,
$\mathcal Q_\omega \mathcal S_{\varphi ,d,\max }^*\mathcal Q_\omega \preceq \mathcal S_{\varphi ,d,\max }$
. In particular,
$$ \begin{align*}\mathcal Q_\omega\mathcal S_{\varphi,d,\max}^*\mathcal Q_\omega K_z(x)=\mathcal S_{\varphi,d,\max}K_z(x)\quad\mbox{for all } z,x\in{\mathbb C}. \end{align*} $$
Lemma 4.1 yields (2b). Hence, by Theorem 4.2, the operator
$\mathcal S_{\varphi ,d,\max }$
is
$\mathcal Q_\omega $
-selfadjoint. Then (2a) follows from
$$ \begin{align*} \mathcal S_{\varphi,d}\preceq \mathcal S_{\varphi,d,\max}=\mathcal Q_\omega\mathcal S_{\varphi,d,\max}^*\mathcal Q_\omega \preceq\mathcal Q_\omega\mathcal S_{\varphi,d}^*\mathcal Q_\omega=\mathcal S_{\varphi,d}.\\[-37pt] \end{align*} $$
Corollary 4.4. Let
$\{\varphi _j:j\in [1,d]_{\mathbb Z}\}$
be a set of distinct, entire functions, that is, condition (1.1) holds. Let
$\mathcal S_{\varphi ,d}$
be the operator arising from the expression
$\mathcal E(\varphi ,d)$
. If the operator
$\mathcal S_{\varphi ,d}$
is real symmetric, then it is
$\mathcal Q_\omega $
-selfadjoint for every
$\omega \in \mathbb {T}$
.
Acknowledgements
The paper was completed during a scientific stay at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank the VIASM for financial support and hospitality.
