1 Introduction

Let \(M_{n} (C)\) be the set of \(n\times n\) complex matrices. For any \(A\in M_{n} (C)\), the conjugate transpose of A is denoted by \(A^{\ast}\). \(A\in M_{n} (C)\) is accretive-dissipative if it has the Hermitian decomposition

$$ A=B+iC,\qquad B=B^{\ast},\qquad C=C^{\ast}, $$
(1.1)

where both matrices B and C are positive definite. Conformally partition A, B, C as

$$ \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} {A_{11} } & {A_{12} } \\ {A_{21} } & {A_{22} } \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} {B_{11} } & {B_{12} } \\ {B_{12}^{\ast}} & {B_{22} } \end{array}\displaystyle \right )+i\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} {C_{11} } & {C_{12} } \\ {C_{12}^{\ast}} & {C_{22} } \end{array}\displaystyle \right ), $$
(1.2)

such that all diagonal blocks are square. Say k and l (\(k,l>0\) and \(k+l=n\)) the order of \(A_{11}\) and \(A_{22}\), respectively, and let \(m=\min\{ k,l\}\). In this article, we always partition A as in (1.2).

If \(B=I_{n} \) in (1.1), then an accretive-dissipative matrix \(A\in M_{n} (C)\) is called a Buckley matrix.

Let \(A=\bigl({\scriptsize\begin{matrix} {A_{11} } & {A_{12} } \cr {A_{21} } & {A_{22} }\end{matrix}} \bigr)\in M_{n} (C)\). If \(A_{11} \) is invertible, then the Schur complement of \(A_{11} \) in A is denoted by \(A/A_{11} :=A_{22} -A_{21} A_{{11}}^{-1} A_{12} \). For a nonsingular matrix A, its condition number is denoted by \(k(A):=\sqrt{\frac{\lambda_{\mathrm{max}} (A^{\ast}A)}{\lambda _{\mathrm{min}} (A^{\ast}A)}} \), which is the ratio of the largest and the smallest singular value of A. For Hermitian matrices \(B,C\in M_{n} (C)\), we write \(B>(\ge )\, C\) to mean that \(B-C\) is Hermitian positive (semi)definite.

If \(A\in M_{n} (C)\) is positive definite, then the famous Fischer-type determinantal inequality ([1], p.478) states that

$$ \det A\le\det A_{11} \cdot\det A_{22} . $$
(1.3)

If \(A\in M_{n} (C)\) is accretive-dissipative, Ikramov [2] first proved the determinantal inequality

$$ \vert {\det A} \vert \le3^{m}\vert {\det A_{11} } \vert \cdot \vert {\det A_{22} } \vert . $$
(1.4)

If \(A\in M_{n} (C)\) is accretive-dissipative, Lin [3] proved the determinantal inequality

$$ \vert {\det A} \vert \le2^{\frac{3m}{2}}\vert {\det A_{11} } \vert \cdot \vert {\det A_{22} } \vert . $$
(1.5)

Recently, Fu and He ([4], Theorem 1) got a stronger result than (1.5) as follows.

Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2). Then

$$ \vert {\det A} \vert \le2^{\frac{m}{2}} \biggl[ {1+ \biggl( { \frac {1-k}{1+k}} \biggr)^{2}} \biggr]^{m}\vert {\det A_{11} } \vert \cdot \vert {\det A_{22} }\vert , $$
(1.6)

where \(k=\max(k(B),k(C))\).

For Buckley matrices, Ikramov [2] obtained the stronger bound

$$ \vert {\det A} \vert \le\biggl(\frac{1+\sqrt{17} }{4} \biggr)^{m}\vert {\det A_{11} } \vert \cdot \vert {\det A_{22} } \vert . $$
(1.7)

In this paper, we will give refinements of (1.6) and (1.7) in Section 2. Other related studies of the Fischer-type determinantal inequalities for accretive-dissipative matrices can be found in [5–7].

2 Main results

We begin this section with the following lemmas.

Lemma 1

([8], Property 6)

Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2). Then \(A/A_{11}\) is also accretive-dissipative.

Lemma 2

([2], Lemma 1)

Let \(A\in M_{n} (C)\) be accretive-dissipative as in (1.1). Then

$$A^{-1}=E-iF, \qquad E=\bigl(B+CB^{-1}C\bigr)^{-1},\qquad F= \bigl(C+BC^{-1}B\bigr)^{-1}. $$

Lemma 3

([9], Lemma 3.2)

Let \(B,C\in M_{n} (C)\) be Hermitian and assume B is positive definite. Then

$$B+CB^{-1}C\ge2C. $$

Lemma 4

([10], (6))

Let \(B=\bigl( {\scriptsize\begin{matrix} {B_{11} } & {B_{12} }\cr {B_{12}^{\ast}} & {B_{22} }\end{matrix}} \bigr)\) be Hermitian positive definite. Then

$$B_{12}^{\ast}B_{11}^{-1} B_{12} \le \biggl( {\frac{1-k(B)}{1+k(B)}} \biggr)^{2}B_{22} . $$

Lemma 5

([3], Lemma 6)

Let \(B,C\in M_{n} (C)\) be positive semidefinite. Then

$$\bigl\vert {\det(B+iC)} \bigr\vert \le\det(B+C). $$

Lemma 6

([11], (1.2))

Let \(a,b>0\). Then

$$\biggl[1+\frac{(\ln a-\ln b)^{2}}{8}\biggr]\sqrt{ab} \le\frac{a+b}{2}. $$

Lemma 7

Let \(B,C\in M_{n} (C)\) be positive definite. Then

$$\det(B+C)\le r^{n}\bigl\vert {\det(B+iC)} \bigr\vert , $$

where \(r= \displaystyle{\max_{1\le j\le n}} \textstyle{\{\sqrt{1+\frac{2}{2+(\ln \lambda_{j} )^{2}}} \}}\), \(\lambda_{j} \) are the eigenvalues of \(B^{-1/2}CB^{-1/2}\), and \(B^{1/2}\) means the unique positive definite square root of B.

Proof

Letting \(a=\lambda_{j} \), \(b=\frac{1}{a} \) in Lemma 6 gives \(1+\lambda_{j}\le\sqrt{1+\frac{2}{2+(\ln\lambda_{j})^{2}}} \vert {1+i\lambda_{j} } \vert \), \(j = 1, \ldots,n\). Then

$$\begin{aligned} \det(B+C) =& \det B\cdot\det\bigl(I+B^{-1/2}CB^{-1/2}\bigr) \\ =& \det B\cdot\prod_{j=1}^{n} {(1+ \lambda_{j} )} \\ \le& \det B\cdot\prod_{j=1}^{n} {\biggl( \sqrt{1+\frac{2}{2+(\ln\lambda_{j} )^{2}}} \vert {1+i\lambda_{j} } \vert \biggr)} \\ \le& \det B\cdot\prod_{j=1}^{n} {\bigl(r \vert {1+i\lambda_{j} } \vert \bigr)} \\ =& r^{n}\det B\cdot\bigl\vert {\det\bigl(I+iB^{-1/2}CB^{-1/2} \bigr)} \bigr\vert \\ =& r^{n}\bigl\vert {\det(B+iC)} \bigr\vert . \end{aligned}$$

This completes the proof. □

Theorem 1

Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2). Then

$$ \vert {\det A} \vert \le \biggl[ {1+ \biggl( { \frac{1-k}{1+k}} \biggr)^{2}} \biggr]^{m}r^{m} \vert {\det A_{11} } \vert \cdot \vert {\det A_{22} } \vert , $$
(2.1)

where \(r= \displaystyle{\max_{1\le j\le n}} \textstyle{\{\sqrt{1+\frac{2}{2+(\ln \lambda_{j} )^{2}}} \}}\), \(\lambda_{j} \) are the eigenvalues of \(B^{-1/2}CB^{-1/2}\), \(B^{1/2}\) means the unique positive definite square root of B, and \(k=\max(k(B),k(C))\).

Proof

By Lemma 2 and Lemma 3, we have

$$\begin{aligned} A/A_{11} =& A_{22} -A_{21} A_{11}^{-1} A_{12} \\ =& B_{22} +iC_{22} -\bigl(B_{12}^{\ast}+iC_{12}^{\ast}\bigr) (B_{11} +iC_{11} )^{-1}(B_{12} +iC_{12} ) \\ =& B_{22} +iC_{22} -\bigl(B_{12}^{\ast}+iC_{12}^{\ast}\bigr) (E_{k} -iF_{k} ) (B_{12} +iC_{12}) \end{aligned}$$

with

$$ E_{k} =\bigl(B_{11} +C_{11} B_{11}^{-1} C_{11} \bigr)^{-1}\le \frac {1}{2}C_{11}^{-1} , \qquad F_{k} = \bigl(C_{11} +B_{11} C_{11}^{-1} B_{11} \bigr)^{-1}\le\frac {1}{2}B_{11}^{-1} . $$
(2.2)

Set \(A/A_{11} =R+iS\) with \(R=R^{\ast}\) and \(S=S^{\ast}\). By Lemma 1, we obtain

$$\begin{aligned}& R=B_{22} -B_{12}^{\ast}E_{k} B_{12} +C_{12}^{\ast}E_{k} C_{12} -B_{12}^{\ast}F_{k} C_{12} -C_{12}^{\ast}F_{k} B_{12} , \\& S=C_{22} +B_{12}^{\ast}F_{k} B_{12} -C_{12}^{\ast}F_{k} C_{12} -C_{12}^{\ast}E_{k} B_{12} -B_{12}^{\ast}E_{k} C_{12} . \end{aligned}$$

It can be proved that

$$\begin{aligned}& \pm\bigl(B_{12}^{\ast}F_{k} C_{12} +C_{12}^{\ast}F_{k} B_{12} \bigr)\le B_{12}^{\ast}F_{k} B_{12} +C_{12}^{\ast}F_{k} C_{12} , \\& \pm\bigl(C_{12}^{\ast}E_{k} B_{12} +B_{12}^{\ast}E_{k} C_{12} \bigr)\le C_{12}^{\ast}E_{k} C_{12} +B_{12}^{\ast}E_{k} B_{12} . \end{aligned}$$

Thus,

$$ R+S\le B_{22} +2B_{12}^{\ast}F_{k} B_{12} +C_{22} +2C_{12}^{\ast}E_{k} C_{12} . $$
(2.3)

As B, C are positive definite, by Lemma 4, we have

$$ B_{12}^{\ast}B_{11}^{-1} B_{12} \le \biggl( {\frac{1-k(B)}{1+k(B)}} \biggr)^{2}B_{22} , \qquad C_{12}^{\ast}C_{11}^{-1} C_{12} \le \biggl( {\frac{1-k(C)}{1+k(C)}} \biggr)^{2}C_{22} . $$
(2.4)

Without loss of generality, we assume \(m=l\), then

$$\begin{aligned} \bigl\vert {\det(A/A_{11} )} \bigr\vert =& \bigl\vert { \det(R+iS)} \bigr\vert \\ \le& \det(R+S) \quad (\mbox{by Lemma 5}) \\ \le& \det\bigl(B_{22} +2B_{12}^{\ast}F_{k} B_{12} +C_{22} +2C_{12}^{\ast}E_{k} C_{12}\bigr) \quad (\mbox{by (2.3)}) \\ \le& \det\bigl(B_{22} +B_{12}^{\ast}B_{11}^{-1} B_{12} +C_{22} +C_{12}^{\ast}C_{11}^{-1} C_{12} \bigr) \quad (\mbox{by (2.2)}) \\ \le& \det \biggl\{ { \biggl[ {1+ \biggl( {\frac{1-k(B)}{1+k(B)}} \biggr)^{2}} \biggr]B_{22} + \biggl[ {1+ \biggl( { \frac{1-k(C)}{1+k(C)}} \biggr)^{2}} \biggr]C_{22} } \biggr\} \quad (\mbox{by (2.4)}) \\ \le& \biggl[ {1+ \biggl( {\frac{1-k}{1+k}} \biggr)^{2}} \biggr]^{m}\det ( {B_{22} +C_{22} } ) \\ \le& \biggl[ {1+ \biggl( {\frac{1-k}{1+k}} \biggr)^{2}} \biggr]^{m}r^{m}\bigl\vert {\det ( {B_{22} +iC_{22} } )} \bigr\vert \quad (\mbox{by Lemma 7}) \\ =& \biggl[ {1+ \biggl( {\frac{1-k}{1+k}} \biggr)^{2}} \biggr]^{m}r^{m}\vert {\det A_{22} } \vert , \end{aligned}$$

where \(k=\max(k(B),k(C))\).

The proof is completed by noting \(\vert {\det A} \vert =\vert {\det A_{11} } \vert \cdot \vert {\det(A/A_{11} )} \vert \). □

Remark 1

Because of \(r\le\sqrt{2}\), inequality (2.1) is a refinement of inequality (1.6).

Theorem 2

Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2) with \(B_{12} =0\). Then

$$ \vert {\det A} \vert \le \biggl( {\frac{\sqrt{17} +1}{4}} \biggr)^{m}\vert {\det A_{11} } \vert \cdot \vert {\det A_{22} } \vert . $$
(2.5)

Proof

Compute

$$\begin{aligned} \vert {\det A} \vert =& \bigl\vert {\det(B+iC)} \bigr\vert \\ =& \det B\cdot\bigl\vert {\det\bigl(I+iB^{-1/2}CB^{-1/2}\bigr)} \bigr\vert \\ \le& \biggl( {\frac{\sqrt{17} +1}{4}} \biggr)^{m}\det B\cdot\bigl\vert {\det\bigl(I_{k} +iB_{{11}}^{-1/2} C_{11} B_{{11}}^{-1/2} \bigr)} \bigr\vert \\ &{}\cdot\bigl\vert {\det \bigl(I_{l} +iB_{{22}}^{-1/2} C_{22} B_{{22}}^{-1/2} \bigr)} \bigr\vert \quad (\mbox{by (1.7)}) \\ =& \biggl( {\frac{\sqrt{17} +1}{4}} \biggr)^{m}\bigl\vert { \det(B_{11} +iC_{11} )} \bigr\vert \cdot\bigl\vert { \det(B_{22} +iC_{22} )} \bigr\vert \\ =& \biggl( {\frac{\sqrt{17} +1}{4}} \biggr)^{m}\vert {\det A_{11} } \vert \cdot \vert {\det A_{22} } \vert . \end{aligned}$$

This completes the proof. □

Remark 2

It is clear that inequality (2.5) is an extension of inequality (1.7).