New results for the non-oscillatory asymptotic behavior of high order differential equations of Poincaré type

1.   Introduction

The study of operator equations has a long history. Khatri and Mitra ^[10], Wu and Cain ^[15], Xiong and Qin ^[16] and Yuan et al. ^[18] studied matrix equations $AX = C$ and $XB = C$ in matrix algebra. Dajić and Koliha ^[1], Douglas ^[4] and Liang and Deng ^[12] investigated these equations in operator space. Xu et al. ^[5,13,14,17], Fang and Yu ^[6] studied these equations of adjoint operator space on Hilbert $C^*$-modules. The famous Douglas' range inclusion theorem played a key role in the existence of the solutions of equation $AX = C$. Many scholars discussed the existence and the general formulae of self-adjoint solutions (resp. positive or real positive solutions) of one equation or common solutions of two equations. In finite dimensional case, Gro$\beta$ gave the necessary and sufficient conditions for the matrix equation $AX = C$ to have a real positive solution in ^[7]. However, the detailed formula for the real positive solution of this equation is not fully provided. Dajíc and Koliha ^[2] first provided a general form of real positive solutions of equation $AX = C$ in Hilbert space under certain conditions. Recently, the real positive solutions of $AX = C$ were considered with corresponding operator $A$ not necessarily having closed range in ^[6,12] for adjoint operators. However, these formulae of real positive solutions still have some additional restrictions. In ^[16], the authors considered the equivalent conditions for the existence of common real positive solutions to the pair of the matrix equations $AX = C$, $XB = D$ in matrix algebra and offered partial common real positive solutions.

Let $\mathcal{H}$ and $\mathcal{K}$ be complex Hilbert spaces. We denote the set of all bounded linear operators from $\mathcal{H}$ into $\mathcal{K}$ by ${\mathcal {B(H, K)}}$ and by ${\mathcal {B(H)}}$ when $\mathcal{H} = \mathcal{K}$. For an operator $A$, we shall denote by $A^*$, $R(A)$, $\overline{R(A)}$ and $N(A)$ the adjoint, the range, the closure of the range and the null space of $A$, respectively. An operator $A\in \mathcal{B(H)}$ is said to be positive ($A\ge 0$), if $\langle Ax, x\rangle\geq 0$ for all $x\in { \mathcal{H}}$. Note that a positive operator has unique square root and

is the absolute value of $A$. Let $A^{\dagger}$ be the Moore-Penrose inverse of $A$, which is bounded if and only if $R(A)$ is closed ^[9]. An operator $A$ is called real positive if

The set of all real positive operators in ${\mathcal B}({\mathcal H})$ is denoted by ${Re}{\mathcal B}_+({\mathcal H})$. An operator sequence $\{T_n\}$ convergents to $T\in {\mathcal B}({\mathcal H})$ in strong operator topology if $||T_nx-Tx||\rightarrow 0$ as $n\rightarrow \infty$ for any $x\in \mathcal H$. Denote by

Let $P_M$ denote the orthogonal projection on the closed subspace $M$. The identity onto a closed subspace $M$ is denoted by $I_M$ or $I$, if there is no confusion.

In this paper, we focus our work on the problem of characterizing the real positive solutions of operator equations $AX = C$ and $XB = D$ with corresponding operators $A$ and $B$ not necessarily having closed ranges in infinite dimensional Hilbert space. Our current goal is three-fold:

Firstly, by using polar decomposition and the strong operator convergence, a completely new representation of the reduced solution $F$ of operator equation $AX = C$ is given by

where $A = U_A |A|$ is the polar decomposition of $A$. This solution has property $F = A^† C$ when $R(A)$ is closed. Furthermore, the necessary and sufficient conditions for the existence of real positive solutions and the detailed formulae for the real positive solutions of equation $AX = C$ are obtained, which improves the related results in ^[2,7,12]. Some comments on the reduced solution and real positive solutions are given.

Next, we discuss common solutions of the system of two operator equations $AX = C$ and $XB = D$. The necessary and sufficient conditions for the existence common solutions and the detailed representations of general common solutions are provided, which extends the classical closed range case with a short proof. Furthermore, we consider the problem of finding the sufficient conditions which ensure this system has a real positive solution as well as the formula of these real positive solutions.

Finally, two examples are provided. As shown by Example 4.1, the system of equations $AX = C$ and $XB = D$ has common real positive solutions under above given sufficient conditions. It is shown by Example 4.2 that a gap is unfortunately contained in the original paper ^[16], where the authors gave two equivalent conditions for the existence of common real positive solutions for this system in matrix algebra. Here, it is still an open question to give an equivalent condition for the existence of common real positive solutions of $AX = C$ and $XB = D$ in infinite dimensional Hilbert space.

2.   The reduced solution of $AX=C$

In general, $A^{\dagger}C$ is the reduced solution of equation $AX = C$ if $R(A)$ is closed. Liang and Deng gave an expression of the reduced solution denoted $A^{‡}C$ through the spectral measure of positive operators where operator $A$ may not be closed, but sometimes $A^{‡}$ is only a symbol since the spectral integral $\int_0^{\infty}\lambda ^{‡}dE_{\lambda}$ (see ^[12]) may be divergent. In this section, we will give a new representation of reduced solution of $AX = C$ by operator sequence. we begin with some lemmas.

Lemma 2.1. (^[8]) Suppose that $\{T_n\}_{n = 1}^\infty$ is an invertible positive operator sequence in ${\mathcal B}({\mathcal H})$ and

If $T$ is also invertible and positive, then

For an operator $T\in {\mathcal B}({\mathcal H})$, then $T$ has a unique polar decomposition $T = U_T |T|$, where $U_T$ is a partial isometry and $P_{\overline{R(T^*)}} = U_T^*U_T$(^[8]). Denote

for each positive integer $n$. In ^[11], the authors paid attention to the operator sequence $T_n|T|$ over Hilbert $C^*$-module. It was given that $T_n|T|$ converges to $P_{\overline{R(T^*)}}$ in strong operator topology for $T$ is a positive operator. Here we have some relative results as follows,

Lemma 2.2. Suppose that $T\in \mathcal{B}(\mathcal H)$ and $T_n = (\frac{1}{n} I_{\mathcal H} + |T|)^{-1}$. Then the following statements hold:

${\rm (i)}$(^[8,11]) $P_{\overline{{R}(T^*)}} = s.o.-lim_{n \rightarrow \infty} T_n|T|$.

${\rm (ii)}$ $T^{\dagger} = s.o.-lim_{n \rightarrow \infty} T_nU_{T}^*$ if $R(T)$ is closed.

Proof. We only prove the statement (ii). If $R(T)$ is closed, then $R(T^*)$ is also closed. It is natural that ${\mathcal H} = R(T^*)\oplus N(T) = R(T)\oplus N(T^*)$. Thus $T$ and $|T|$ have the following matrix forms, respectively,

and

where $T_{11}\in {\mathcal B}(R(T^*), R(T))$ is invertible. Then $T^†$ has the matrix form

And from (2.2), it is easy to get that

By the invertiblity of diagonal operator matrix and the above matrix form (2.4), we have

Because the operator sequence

converges to $|T_{11}|$ and

is invertible in ${\mathcal B}(R(T^*))$ for each $n$, then the operator sequence

converges to $|T_{11}|^{-1}$ by Lemma 2.1. Moreover, partial isometry $U_T$ has the matrix form

with respect to the space decomposition ${\mathcal H} = R(T^*)\oplus N(T)$, where $U_{11}$ is a unitary from $R(T^*)$ onto $R(T)$. Then $T_{11} = U_{11}|T_{11}|$ and $T_{11}^{-1} = |T_{11}|^{-1} U_{11}^*$ by $T = U_T|T|$ and the matrix forms (2.1), (2.2), and (2.6). From matrix forms (2.5) and (2.6), we have

Since the operator sequence $(\frac{1}{n}I_{R(T^*)}+ |T_{11}|)^{-1}U_{11}^*$ converges to

we obtain that the operator sequence $\{T_nU_T^*\}$ converges to $\left(\begin{array}{cc} T_{11}^{-1} & 0 \\ 0 & 0 \\ \end{array}\right)$, which is equivalent to $T^{\dagger}$ by the matrix form (2.3).□

In Lemma 2.2, the statement (ii) is not necessarily true if $R(T)$ is not closed. The following is an example.

Example 2.3. Let $\{e_1, e_2, \cdots\}$ be a basis of an infinite Hilbert space $\mathcal H$. Define an operator $T$ as follows,

Then

since $T$ is a positive operator. It is easy to get that

for any $k$. Suppose

Then $x\in \mathcal H$ with

By direct computing, we have

that is, $\|T_nx \| > \frac{1}{\|x\|}\left(\sum_{k = 1}^{n} \frac{1 }{k }-\frac{1}{2}\right).$ It is shown that the number sequence $\{\|T_nx\|\}$ is divergent since the harmonic progression $\sum_{k = 1}^{\infty} \frac{1 }{k }$ is divergent. This implies that the operator sequence $\{T_n\}$ is divergent in strong operator topology.

Lemma 2.4. (^[4]) Let $A, C \in {\mathcal B}({\mathcal H})$. The following three conditions are equivalent:

${\rm (1)}$ ${\mathcal R}(C) \subseteq \mathcal{R}(A)$.

${\rm (2)}$ There exists $\lambda > 0$ such that $CC^* \leq \lambda AA^*$.

${\rm (3)}$ There exists $D\in{\mathcal B}({\mathcal H})$ such that $AD = C$.

If one of these conditions holds then there exists a unique solution $F\in {\mathcal B}({\mathcal H})$ of the equation $AX = C$ such that $R(F)\subseteq R(A^*)$ and $N(F) = N(C)$. This solution is called the reduced solution.

Theorem 2.5. Let $A, C\in{\mathcal B}({\mathcal H})$ be such that the equation $AX = C$ has a solution. Then the unique reduced solution $F$ can be formulated by

where $U_A\in\mathcal{B}(H)$ is the partial isometry associated to the polar decomposition $A = U_A |A|$. Particularly, if $R(A)$ is closed, the reduced solution $F$ is $A^† C$.

Proof. Since $A = U_A |A|$, we have $U_A ^* A = |A|$. Assuming that $X$ is a solution of $AX = C$, then $|A|X = U_A^*C$. By multiplication $(|A|+\frac{1}{n}I)^{-1}$ on the left, it follows that

From Lemma 2.2 (i), $\{(|A|+\frac{1}{n})^{-1}|A|X\}$ is convergent to $P_{\overline{R(A^*)}}X$ in strong operator topology. So

This shows that

$R(F)\subseteq\overline{R(A^*)}$ and $N(F)\subseteq N(C)$. By the definition of $F$, we know $N(C)\subseteq N(F)$. So $N(C) = N(F)$. Suppose that $F^{\prime}$ is another reduced solution, then $A(F-F^{\prime}) = 0$ and so

It is shown that $F-F^{\prime} = 0$ and then $F = F^{\prime}.$ That is to say, $F$ is the unique reduced solution of $AX = C$. If $R(A)$ is closed, $F = A^{\dagger }C$ by Lemma 2.2 (ii).□

Remark 2.6. Although $(|A|+\frac{1}{n}I)^{-1}U_{A}^*$ does not necessarily converge in strong operator topology by Example 2.3, $(|A|+\frac{1}{n}I)^{-1}U_{A}^*C$ is convergent from the proof of the Theorem 2.5 if $R(C)\subseteq R(A)$.

As a consequence of the above Theorem 2.5, Lemma 2.4 and Theorem 3.5 in ^[12], the following result holds.

Theorem 2.7. (^[12]) Let $A, C\in {\cal B}({\mathcal H})$. Then $AX = C$ has a solution if and only if $R(C)\subseteq R(A)$. In this case, the general solutions can be represented by

where $F$ is the reduced solution of $AX = C$ and $P = P_{\overline{R(A^*)}}$. Particularly, if $R(A)$ is closed, then the general solutions can be represented by

3.   Real positive solutions of $AX=C, XB=D$

In this section, we mainly study the real positive solutions of equation

and the system of equations

Firstly, some preliminaries are given. For a given operator $A\in {\mathcal B}({\mathcal H})$, denote $P = P_{\overline{R(A^*)}}$ in the sequel.

Lemma 3.1. Let $A, C\in {\mathcal B}({\mathcal H})$ be such that Eq (3.1) has a solution and $F$ is the reduced solution. Then the following statements hold:

${\rm (i)}$ $CA^*+AC^*\geq 0$ if and only if $FP+P F^*\geq 0$.

${\rm (ii)}$$R(FP+PF^*)$ is closed if $R(CA^*+AC^*)$ is closed.

Proof. (i) Assume that $CA^*+AC^*\geq 0$. For any $x\in \mathcal H$, there exist $y\in \overline{R(A^*)}$ and $z\in N(A)$ such that $x = y+z$. And there exists $\{A^*y_n\}\subset R(A^*)$ such that $lim_{n\rightarrow \infty}A^*y_n = y$. According to $R(F)\subset \overline{R(A^*)}$, by direct computing,

This shows that $FP+PF^*\geq 0$.

Contrarily, if $FP+PF^*\geq 0$, it is elementary to get that

for any $x\in \mathcal H$ by $R(F)\subset \overline{R(A^*)}$. This implies that $CA^*+AC^* \geq 0$.

(ii) Since $R(CA^*+AC^*) = R(A(F+F^*)A^*)$ is closed, we have $R(A(F+F^*)P)$ is closed. It follows that $R(P(F+F^*)A^*)$ is closed and so $R(P(F+F^*)P)$ is closed.□

Lemma 3.2. (^[3]) Let

The following two statements hold:

${\rm (i)}$ There exists an operator $A_{22}$ such that $A\geq 0$ if and only if $A_{11}\geq 0$ and $R(A_{12})\subseteq R(A_{11}^{\frac{1}{2}})$.

${\rm (ii)}$ Suppose that $A_{11}$ is an invertible and positive operator. Then $A\geq 0$ if and only if $A_{22}\geq A_{12}^* A_{11}^{-1} A_{12}$.

Lemma 3.3. Let

Then $A\geq 0$ if and only if

${\rm (i)}$ $A_{11}\geq 0$.

${\rm (ii)}$$A_{22}\geq X^* P_{\overline{R(A_{11}})}X$, where $X$ is a solution of $A_{11}^{\frac{1}{2}}X = A_{12}$.

Proof. Assume that the statements (i) and (ii) hold. Then

Conversely, if $A\geq 0$, it follows from Lemma 3.2 that $A_{11}\geq 0$. And also we get that

has a solution $X$ since $R(A_{12})\subset R(A_{11}^{\frac{1}{2}})$. Moreover, $A\geq 0$ shows that for any positive integer $n$,

that is,

By Lemma 3.2 again, we have

From Lemma 2.2 (i), it is immediate that

in strong operator topology. Thus

holds. Therefore $A_{22}\geq X^* P_{\overline{R(A_{11}})}X.$□

Corollary 3.4. Let

be such that $A_{11}$ has closed range. Then $A\geq 0$ if and only if

${\rm (i)}$ $A_{11}\geq 0$.

${\rm (ii)}$$A_{22}\geq X^* A_{11} X$, where $X$ is a solution of $A_{11}X = A_{12}$.

Proof. Since $A_{11}$ is an operator with closed range, then $R(A_{11}^{\frac{1}{2}}) = R(A_{11})$. Thus the solvability of the two equations $A_{11} X = A_{12}$ and $A_{11}^{\frac{1}{2}} X = A_{12}$ are equivalent. It is easy to check that $A_{11}^{\frac{1}{2}} X$ is a solution of $A_{11}^{\frac{1}{2}} X = A_{12}$ if $X$ is a solution of $A_{11}X = A_{12}$. From the Lemma 3.3, the result holds.□

In ^[12], the author gave a formula of real positive solutions for Eq (3.1) which is only a restriction on the general solutions. Here we obtain a specific expression of the real positive solutions.

Theorem 3.5. Let $A, C\in {\cal B}({\mathcal H})$. The Eq (3.1) has real positive solutions if and only if $R(C)\subseteq R(A)$ and $CA^*+AC^*\geq 0$. In this case, the real positive solutions can be represented by

for any $Y\in {\mathcal B}({\mathcal H})$, $Z\in Re{\mathcal B}_+({\mathcal H})$, where $F$ is the reduced solution of $AX = C$ and $F_0 = PF^*+FP$. Particularly, if $R(CA^*+AC^*)$ is closed, then the real positive solutions can be represented by

Proof. From Theorem 2.7, $AX = C$ has a solution if and only if $R(C)\subseteq R(A)$. Suppose that $X$ is a solution of $AX = C$. There exists an operator $\overline{Y}\in {\cal B}({\mathcal H})$ such that

Set ${\mathcal H} = \overline {R(A^*)}\oplus N(A)$. Then $F$ has the following matrix form

since $R(F)\subset \overline{R(A^*)}$. The operator $\overline{Y}$ can be expressed as

This follows that

"$\Rightarrow$": If $X$ is a real positive solution, then $F_{11} +F_{11}^* \geq 0$ from Lemma 3.3 and the matrix form (3.7) of $X+X^*$. And so $PF^*+FP\geq 0$. According to Lemma 3.1, $CA^*+AC^*\geq 0$.

"$\Leftarrow$": Assume that $CA^*+AC^*\geq 0$, then $PF^*+FP\geq 0$ and so $F_{11} +{F_{11}}^* \geq 0$. Denote $X_0 = F-(I-P)F^*$. Then $AX_0 = C$ and

That is, $X_0$ is a real positive solution of $AX = C$.

Next, we analyse the general form of real positive solutions. Suppose that $X$ is a real positive solution of $AX = C$. Then $X+X^*$ has the matrix form (3.7). From Lemma 3.3, there exists an operator $Y_{12}$ such that

and

Let

In this case, $X$ can be represented by the following form

for some $Y\in {\mathcal B}({\mathcal H})$, $Z\in Re{\mathcal B}_+({\mathcal H})$, where $F_{0} = PF^*+FP$.

On the contrary, for any operator $Y\in {\mathcal B}(\mathcal H)$ and $Z\in Re{\mathcal B}_+(\mathcal H)$ such that $X$ has the form (3.3), it is clear that $AX = C$. Let $Y = (Y_{ij})_{2\times 2}$ and $Z = (Z_{ij})_{2\times 2}$ with respect to the space decomposition $\mathcal H = \overline{R(A^*)}\oplus N(A)$. Then

Particularly, if $R(CA^*+AC^*)$ is closed, then $R(F_{11}+F_{11}^*)$ is closed by Lemma 3.1. Suppose that $X$ is a real positive solution of $AX = C$, then $X+X^*$ has the matrix form (3.7). From Corollary 3.4, there exists an operator $Y_{12}\in {\mathcal B}(N(A), \overline{R(A^*)})$ such that

and

Let

Consequently,

for some $Y\in {\mathcal B}({\mathcal H})$, $Z\in Re{\mathcal B}_+({\mathcal H})$.

On the contrary, for any $Y\in {\mathcal B}(\mathcal H)$ and $Z\in Re{\mathcal B}_+(\mathcal H)$ such that $X$ has a form (3.2), it is clear that $AX = C$ and also we have

□

Next, we consider the solutions of $AX = C, XB = D$.

Theorem 3.6. Let $A, B, C, D\in {\cal B}({\mathcal H})$. Then the system of Eq (3.2) has a solution if and only if $R(C)\subseteq R(A)$, $R(D^*)\subseteq R(B^*)$ and $AD = CB$. In this case, the general common solution can be represented by

for any $Z\in {\cal B}({\mathcal H})$, where $F$ is the reduced solution of (3.1) and $H$ is the reduced solution of $B^*X = D^*$. Particularly, if $R(A)$ and $R(B)$ are closed, the general solution can be represented by

Proof. The necessity is clear. We only need to prove the sufficient condition. From $R(C)\subseteq R(A)$ and Theorem 2.7, a solution $X$ of Eq (3.1) can be represented by

for some $Y\in {\cal B}({\mathcal H})$, where $F$ is the reduced solution. Substitute above $X$ into equation $XB = D$, we have

This shows that

and then

since $R(F)\subseteq \overline{R(A^*)}$ and $(I-P)F = 0$. Moreover, $YB = D$ has a solution since $R(D^*)\subseteq R(B^*)$. Denote $H$ is the reduced solution of $B^*\hat{Y} = D^*$. Then

and

since $R(H)\subseteq \overline{R(B)}$ and $N(H) = N(D^*)$. So $H(I-P)$ is the reduced solution of $B^* \hat{Y} = D^*(I-P)$ and then there exists $Z \in {\mathcal B}({\mathcal H})$ such that

That is to say, the system of equations $AX = C$, $XB = D$ has common solutions. The general solution can be represented by

Especially, if $R(A)$ and $R(B)$ are closed, then $F = A^† C$ and $H = {B^*}^† D^*$. It follows that the general common solution is

for any $Z\in {\cal B}({\mathcal L}, {\mathcal H})$.□

Theorem 3.7. Let $A, B, C, D\in {\cal B}({\mathcal H})$ and $Q = P_{\overline{R((I-P)B)}}$. The system of Eq (3.2) has a real positive solution if the following statements hold:

${\rm (i)}$$R(C)\subseteq R(A)$, $R(D^*)\subseteq R(B^*)$, $AD = CB$,

${\rm (ii)}$ $CA^*$, $(D^*+B^*F)(I-P)B$ are real positive operators,

${\rm (iii)}$ $R((D^*+B^*F)(I-P))\subseteq R(B^*(I-P))$,

where $F$ is the reduced solution of $AX = C$. In this case, one of common real positive solutions can be represented by

for any $Z\in Re{\cal B}_+({\mathcal H})$, where $H$ is the reduced solution of

Proof. Combined Theorems 3.5 and 3.6 with statements (i) and (ii), the system of equations $AX = C$, $XB = D$ has common solutions. For any $Y\in Re{\mathcal B}_+({\mathcal H})$,

is a real positive solution of $AX = C$. We only need to prove that there exists $Y\in Re{\mathcal B}_+({\mathcal H})$ such that above $X$ is also a solution of $XB = D$. Since

and also

then $A$, $F$ and $B$ have the following operator matrix forms,

respectively, where $A_{11}$ is an injective operator from $\overline{R(A^*)}$ into $\overline{R(A)}$ with dense range. The operator $D$ has the matrix form

since $R(D^*)\subseteq R(B^*)$. Let

Then $X$ has a matrix form as follows:

By the matrix forms (3.11) and (3.14) of $B$ and $X$, respectively, it is easy to get that

Combining $AD = CB = AFB$ with the matrix forms (3.9)–(3.12), we have

It is immediate that

And so

Therefore,

since $R(A_{11}^*)$ has dense range in $\overline{R(A_{11}^*)}$. From the statement (iii), $R(D_{21}^* +B_{11}^* F_{12})\subseteq R(B_{21}^*)$ holds. Moreover, by the statement (ii) $(D^*+B^*F)(I-P)B$ being real positive, we have $(D_{21}^* +B_{11}^* F_{12})$ is real positive. These infer that the equation

has real positive solutions by Theorem 3.5. Suppose that $H_{22}$ is the reduced solution of Eq (3.17), we can deduce that

is the reduced solution of the following equation

since $R(H)\subseteq(I-P)B$ and

Using Theorem 3.5 again, $H-(I-Q)H^*\geq 0$ is a real positive solution of (3.18). Then $(I-P)(H-(I-Q)H^*)(I-P)$ is also a real positive solution of (3.18). Because of

we have

Let

That is,

Put $X = X_0$ into formula (3.15), combining Eqs (3.16) and (3.17) with the matrix form (3.12) and (3.15), we can get $B^*X_0^* = D^*$. So $X_0$ is a common real positive solution of Eq (3.2). Furthermore, it is easy to verify that

and

for any $Z\in {\mathcal B}(\mathcal H)$. Therefore

is a real positive solution of system equations $AX = C, XB = D$.□

4.   Examples

Some examples are given in this section to demonstrate our results are valid. And also it is shown that a gap is unfortunately contained in the original paper ^[16] about existence of real positive solutions.

Example 4.1. Let $\mathcal H$ be the infinite Hilbert space as in Example 2.3 with a basis $\{e_1, e_2, \cdots\}$ and $Te_k = \frac{1}{k}e_{k}, \forall k \geq 1.$ As well known the range of $T\in {\mathcal B}(\mathcal H)$ is not closed and $\overline{R(T)} = \mathcal H$. Define operators $A, C, B, D\in {\mathcal B}(\mathcal H \oplus \mathcal H$ as follows:

Then

and

By Theorem 2.5 and Lemma 2.2 (i), the reduced solution $F$ of the operator equation $AX = C$ has the following form

Denote

From Theorem 3.5, we know that the formula of real positive solutions of $AX = C$ is

Furthermore, it is easy to check that

${\rm (i)}$ $AD = CB, R(C) = R(A), R(D^*) = R(B^*)$.

${\rm (ii)}$ $CA^* = A$ and $(D^*+B^*F)(I-P)B = \left(\begin{array}{cccc} 0 & 0\\ 0 & 0 \\ \end{array}\right)$ are real positive operators.

${\rm (iii)}$$(D^*+B^*F)(I-P) = \left(\begin{array}{cccc} 0 & 0\\ 0 & 0 \\ \end{array}\right)$ and $B^*(I-P) = \left(\begin{array}{cccc} 0 & 0\\ 0&T \\ \end{array}\right)$.

By Theorem 3.7, we have the system of equations

has common real positive solutions. Moreover,

So one of real positive solutions has the following form

Whereas the statements (i)–(iii) in Theorem 3.7 are only sufficient conditions for the existence of common real positive solutions of system (3.2). The following is an example.

Example 4.2. Let $A, B, C, D$ be $2\times 2$ complex matrices. Denote

By direct computing, $AX_0 = C$, $X_0B = D$ and

So $X_0$ is a common real positive solution of system equations $AX = C, XB = D$. But in this case, Theorem 3.7 does not work. In fact,

Hence,

and

This shows that the statement (iii)

in Theorem 3.7 does not hold.

Xiong and Qin gave two equivalent conditions (Theorems 2.1 and 2.2 in ^[16]) for the existence of common real positive solutions for $AX = C, XB = D$ in matrix algebra. But unfortunately, there is a gap in these results. Here, Example 4.2 is a counterexample. In fact, $r(A) = 1\neq 2$, where $r(A)$ stands for the rank of matrix $A$ and $A = A^* = CA^*$. Simplifying by elementary block matrix operations, we have

Moreover,

Therefore, combining the above result with formula (4.1), we obtain that

Then

Noting that

Using elementary row-column transformation again, we have

From formulaes (4.1), (4.2) and (4.4), it is natural to get that

since $AD = CB$. But $AX = C, XB = D$ have a common real positive solution. The statements (4.3) and (4.5) shows that Theorems 2.1 and 2.2 in ^[16] do not work, respectively. Actually, the conditions in ^[16] are also only sufficient conditions for the existence of common real positive solutions of Eq (3.2). So here is still an open question.

Question 4.3. Let $A, B, C, D\in {\cal B}({\mathcal H})$. Give an equivalent condition for the existence of common real positive solutions of $AX = C, XB = D$.

5.   Conclusions

In this work, a new representation of reduced solution of $AX = C$ is given by a strong operator convergent sequence. This result provides us a method to discuss the general solutions of Eq (3.2). By making full use of block operator matrix methods, the formula of real positive solutions of $AX = C$ is obtained in Theorem 3.5, which is the basis of finding common real positive solutions of Eq (3.2). Through Example 4.1, it is demonstrated that Theorem 3.7 is useful to find some common real positive solutions. But unfortunately, it is complicated to consider all the common real positive solutions by using of the method in Theorem 3.7. Maybe, we need some other techniques. This will be our next problem to solve.

Acknowledgements

The authors would like to thank the referees for their useful comments and suggestions, which greatly improved the presentation of this paper. Part of this work was completed during the first author visit to Shanghai Normal University. The author thanks professor Qingxiang Xu for his discussion and useful suggestion.

This research was supported by the National Natural Science Foundation of China (No. 12061031), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2021JM-189) and the Natural Science Basic Research Plan in Hainan Province of China (Nos. 120MS030,120QN250).

Conflict of interest

The authors declare no conflicts of interest.