1 Introduction
Consider a complex Hilbert space $( H,\langle \cdot \, ,\cdot \rangle ) $ . An operator T is said to be positive (denoted by $ T\geq 0$ ) if $\langle Tx,x\rangle \geq 0$ for all $x\in H$ and strictly positive (denoted by $T>0$ ) if T is positive and invertible. A real valued continuous function $ f $ on $(0,\infty )$ is said to be operator monotone if $f(A)\geq f(B)$ holds for any $A\geq B>0.$ As usual, by $A\geq B$ , we understand that $A-B\geq 0.$
We have the following representation of operator monotone functions (see for instance [Reference Bhatia1, pages 144–145]).
Theorem 1.1 (Löwner, [Reference Löwner5]).
A function $f:[0,\infty )\rightarrow \mathbb {R}$ is operator monotone in $[0,\infty )$ if and only if it has the representation
where $b\geq 0$ and $\mu $ is a positive measure on $(0,\infty )$ such that
A real valued continuous function f on an interval I is said to be operator convex (operator concave) on I if
in the operator order, for all $\lambda \in [ 0,1] $ and for all selfadjoint operators A and B on a Hilbert space H whose spectra are contained in $I.$ Notice that a function f is operator concave if $-f$ is operator convex.
We have the following representation of operator convex functions [Reference Bhatia1, page 147].
Theorem 1.2. A function $f:(0,\infty )\rightarrow \mathbb {R}$ is operator convex in $(0,\infty )$ with $f_{+}^{\prime }( 0) \in \mathbb {R}$ if and only if it has the representation
where $c\geq 0$ and $\mu $ is a positive measure on $(0,\infty )$ such that (1.2) holds.
Assume that A, $B\geq 0$ . Moslehian and Najafi [Reference Moslehian and Najafi6] showed that $AB+BA$ is positive if and only if the operator subadditivity property holds, that is,
for all nonnegative operator monotone functions f on $[0,\infty ).$ For some interesting consequences of this result, see [Reference Moslehian and Najafi6].
We have the following integral representation for the power function when $ t>0$ , $r\in (0,1]$ (see for instance [Reference Bhatia1, page 145]),
Motivated by these representations, we introduce, for a continuous and positive function $w( \lambda ) $ , $\lambda>0$ , the integral transform
where $\mu $ is a positive measure on $(0,\infty )$ and the integral (1.4) exists for all $t>0.$ For $\mu $ , the usual Lebesgue measure, we put
Now, assume that $T>0$ . By the continuous functional calculus for selfadjoint operators, we can define the positive operator
where w and $\mu $ are as above. When $\mu $ is the usual Lebesgue measure, for $T>0$ ,
If we take $\mu $ to be the usual Lebesgue measure and the kernel $w_{r}( \lambda ) =\lambda ^{r-1}$ , $r\in (0,1],$ then
Motivated by these results, we show among other things that if B, $A>0,$ then
that is, $\mathcal {D}( w,\mu ) $ is operator subadditive on $( 0,\infty ) .$ From this, if $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator monotone function on $[0,\infty )$ , we show that the function $[ f( t) -f( 0) ] t^{-1}$ is operator subadditive on $( 0,\infty ) .$ Also, if $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator convex function on $[0,\infty )$ , then the function $[ f( t) -f( 0) -f_{+}^{\prime }( 0) t] t^{-2}$ is operator subadditive on $( 0,\infty ) .$ Some examples for integral transforms $\mathcal {D}( \cdot \, ,\cdot ) $ related to the exponential and logarithmic functions are also provided.
2 Subadditivity property
Theorem 2.1. For all A, $B>0$ ,
namely, $D(w,\mu )$ is operator subadditive.
Proof. For all A, $B>0,$ by using the representation of $D(w,\mu ),$
For $\lambda \geq 0,$ define the operator
If we multiply both sides of $K_{\lambda }$ by $A+B+\lambda ,$ then we obtain successively
By multiplying both sides of (2.3) by $( A+B+\lambda ) ^{-1}$ ,
We then have the representation
for all A, $B>0.$
Since A, $B>0$ and $\lambda \geq 0,$ we obtain $L_{\lambda }\geq 0$ from (2.3). By (2.4), $K_{\lambda }\geq 0$ , and multiplying by $w( \lambda ) \geq 0$ and integrating over the measure $\mu $ gives (2.1).
Corollary 2.2. Assume that $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator monotone function on $[0,\infty )$ . If A, $B>0$ , then
that is, the function $[ f( t) -f( 0) ] t^{-1}$ is operator subadditive on $( 0,\infty ) .$ In particular, if $f( 0) =0,$ then
Proof. If $f:[0,\infty )\rightarrow \mathbb {R}$ is operator monotone, then by (1.1),
for some positive measure $\mu ,$ where $\ell ( \lambda ) =\lambda $ , $\lambda>0.$ By applying Theorem 2.1 for $\mathcal {D}( \ell ,\mu ) $ and performing the required calculations, we deduce
for all A, $B>0,$ which gives
for all A, $B>0$ and (2.5) is obtained.
Remark 2.3. If we take $f( t) =t^{r}$ , $r\in (0,1]$ , in (2.6), then we get the power inequality
for all A, $B>0.$
If we take $f( t) =\ln ( t+1) $ in (2.6), then we get the logarithmic inequality
Similar inequalities can be obtained by using the examples of operator monotone functions from [Reference Fujii and Seo2, Reference Furuta3] and the references therein.
Corollary 2.4. Assume that $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator convex function on $[0,\infty )$ . If A, $B>0,$ then
that is, the function $[ f( t) -f( 0) -f_{+}^{\prime }( 0) t] t^{-2}$ is operator subadditive on $ ( 0,\infty ) .$
If $f( 0) =0,$ then
Proof. If $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator convex function on $ [0,\infty ),$ then by (1.3),
for some positive measure $\mu ,$ where $\ell ( \lambda ) =\lambda $ , $\lambda>0.$
By applying Theorem 2.1 for $\mathcal {D}( \ell ,\mu ) $ and performing the required calculations, we deduce
for all A, $B>0.$ From this,
which proves (2.7).
Remark 2.5. Let $a>0$ and $p\in \lbrack -1,0)\cup \lbrack 1,2].$ Then for all A, $B>0$ , we have the power inequality
We also have the logarithmic inequality
for all A, $B>0.$
3 Reverse inequalities
We define the difference $\mathcal {D}( w,\mu ) ( \cdot \, ,\cdot ) $ for positive numbers t, s by
and the difference for positive operators A, $B,$
for a continuous and positive function $w( \lambda ) $ , $\lambda>0$ and $\mu $ a positive measure on $(0,\infty )$ such that the integral (1.4) exists for all $t\geq 0.$ We prove the following reverse inequality.
Theorem 3.1. Assume that there exists positive constants $\alpha $ , $\beta ,$ $\gamma $ and $\delta $ such that
Then,
Proof. Observe that
and
that is,
which gives
and so
for all $\lambda \geq 0.$ If we multiply by $w( \lambda ) \geq 0$ and integrate, then by (2.2),
which gives
Observe that
and
which implies that
Therefore,
The case of operator monotone functions is the following corollary.
Corollary 3.2. Assume that $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator monotone function on $[0,\infty )$ with $f( 0) =0.$ If $ A$ , $B>0$ satisfy the condition (3.1), then
Proof. Write
for some positive measure $\mu ,$ where $\ell ( \lambda ) =\lambda $ , $\lambda>0$ . From (3.2),
which proves the desired result (3.5).
Remark 3.3. If A, $B>0$ satisfy the condition (3.1) and $r\in (0,1],$ then we have the reverse power inequality
The case of operator convex functions is the following corollary.
Corollary 3.4. Assume that $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator convex function on $[0,\infty )$ with $f( 0) =0.$ If A, $B>0$ satisfy the condition (3.1), then
Proof. The result follows from (3.2) observing, by (1.3), that
for some positive measure $\mu ,$ where $\ell ( \lambda ) =\lambda $ , $\lambda>0.$
Remark 3.5. If A, $B>0$ satisfy the condition (3.1), then by taking $f( t) =-\ln ( t+1)$ in (3.6), we obtain
4 Some examples
We define the upper incomplete Gamma function
which for $z=0$ , gives the Gamma function
We have the integral representation [Reference Paris8]
for $\mbox {Re\,}a<1$ and $\vert \mbox {arg\,}z\vert <\pi .$
Now, we consider the weight $w_{\cdot ^{-a}e^{-\cdot }}( \lambda ) :=\lambda ^{-a}e^{-\lambda }$ for $\lambda>0.$ Then by (4.1),
for $a<1$ and $t>0.$ For $a=0$ in (4.2),
for $t>0,$ where
From Theorem 2.1, we conclude that $\mathcal {D}( w_{\cdot ^{-a}e^{-\cdot }}) $ and, in particular, $\mathcal {D}( w_{e^{-\cdot }}) $ are operator subadditive on $( 0,\infty ) .$
We can also consider the weight $w_{( \cdot ^{2}+a^{2}) ^{-1}}( \lambda ) :={1}/{(\lambda ^{2}+a^{2})}$ for $\lambda>0$ and $a>0.$ Then, by simple calculations,
for $t>0$ and $a>0.$ For $a=1$ , we also have
for $t>0.$ If $T>0$ and $a>0,$ then
and, in particular,
From Theorem 2.1, we conclude that $\mathcal {D}( w_{( \cdot ^{2}+a^{2}) ^{-1}}) $ and, in particular, $\mathcal {D} ( w_{( \cdot ^{2}+1) ^{-1}}) $ are operator subadditive on $( 0,\infty ) .$
Other similar inequalities can be obtained by using the examples of operator monotone functions provided in [Reference Fujii and Seo2–Reference Furuta4, Reference Moslehian and Najafi6, Reference Moslehian and Najafi7].