Operator versions of H\"older inequality and Hilbert $C^*$-modules

TL;DR

This paper explores operator versions of H"older inequalities within Hilbert $C^*$-modules, extending classical inequalities to operator settings and unitarily invariant norms.

Contribution

It introduces new operator H"older inequalities based on recent weighted Cauchy-Schwarz results in Hilbert $C^*$-modules.

Findings

Derived new H"older type inequalities for Hilbert space operators

Extended classical inequalities to the setting of Hilbert $C^*$-modules

Applied inequalities to unitarily invariant norms

Abstract

Recently proved weighted Cauchy Scwarz inequality for Hilbert $C^*$-modules leads to many H\"older type inequalities for unitarily invariant norms on Hilbert space operators.

Equations

$$a♯b=a^{1/2}(a^{-1/2}b{a}^{-1/2}{)}^{1/2}{a}^{1/2},$$

$$|⟨x,Ty⟩|\le u^{\ast }⟨x,f(|T^{\ast }|{)}^{2}x⟩u♯⟨y,g(|T|{)}^{2}y⟩.$$

$$\left|⟨x,y⟩\right|\le u^{\ast }⟨x,x⟩u♯⟨y,y⟩,⟨x,y⟩=u\left|⟨x,y⟩\right|,$$

$$\left|⟨x,Ty⟩\right|=\left|⟨x,U|T|y⟩\right|=\left|⟨f(|T|)U^{\ast }x,g(|T|)y⟩\right|,$$

$$\begin{gathered} \left|⟨x,Ty⟩\right|\le u^{\ast }⟨f(|T|)U^{\ast }x,f(|T|)U^{\ast }x⟩u♯⟨g(|T|)y,g(|T|)y⟩= \\ =u^{\ast }⟨x,Uf(|T|{)}^2{U}^{\ast }x⟩u♯⟨y,g(|T|{)}^{2}y⟩. \end{gathered}$$

$$\left|\left|\left||X{|}^r\right|\right|\right|\le {\left|\left|\left|A^{pr/2}\right|\right|\right|}^{1/p}{\left|\left|\left|B^{qr/2}\right|\right|\right|}^{1/q}.$$

$$\left|\left|\left|{\left|⟨x,Ty⟩\right|}^r\right|\right|\right|\le {\left|\left|\left|{⟨x,f(|T^{\ast }|{)}^{2}x⟩}^{pr/2}\right|\right|\right|}^{1/p}{\left|\left|\left|{⟨y,g(|T|{)}^{2}y⟩}^{qr/2}\right|\right|\right|}^{1/q},$$

$$X=⟨x,f(|T^{\ast }|{)}^{2}x⟩,Y=⟨y,g(|T|{)}^{2}y⟩.$$

$$\left[\begin{array}{cc}{u}^{\ast }Xu& \left|⟨x,Ty⟩\right|\\ \left|⟨x,Ty⟩\right|& Y\end{array}\right]\ge 0.$$

$$0\le \left[\begin{array}{cc}u& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}{u}^{\ast }Xu& \left|⟨x,Ty⟩\right|\\ \left|⟨x,Ty⟩\right|& Y\end{array}\right]\left[\begin{array}{cc}{u}^{\ast }& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}X& ⟨x,Ty⟩\\ {⟨x,Ty⟩}^{\ast }& Y\end{array}\right]$$

$$\left|\left|\left|{\left|{\int }_{\Omega }{A}_t^{\ast }{X}_t{B}_t\mathrm{d}\mu (t)\right|}^r\right|\right|\right|\le {\left|\left|\left|{\left({\int }_{\Omega }{A}_t^{\ast }f(|X_t^{\ast }|{)}^2{A}_t\right)}^{\frac{pr}{2}}\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|{\left({\int }_{\Omega }{B}_t^{\ast }g(|X_t|{)}^2{B}_t\mathrm{d}\mu (t)\right)}^{\frac{qr}{2}}\right|\right|\right|}^{\frac{1}{q}}.$$

$$\left|\left|\left|{\left|{\int }_{\Omega }{A}_t^{\ast }{X}_t{B}_t\mathrm{d}\mu (t)\right|}^r\right|\right|\right|\le {\left|\left|\left|{\left({\int }_{\Omega }{A}_t^{\ast }|X_t^{\ast }{|}^{2\alpha }{A}_t\right)}^{\frac{pr}{2}}\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|{\left({\int }_{\Omega }{B}_t^{\ast }|X_t{|}^{2\beta }{B}_t\mathrm{d}\mu (t)\right)}^{\frac{qr}{2}}\right|\right|\right|}^{\frac{1}{q}},$$

$$\left|\left|\left|{\left|{\int }_{\Omega }{A}_t^{\ast }{X}_t{B}_t\mathrm{d}\mu (t)\right|}^r\right|\right|\right|\le {\left|\left|\left|{\int }_{\Omega }|A_t{|}^{rp}\mathrm{d}\mu (t)\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|{\int }_{\Omega }|B_t{|}^{rq}\mathrm{d}\mu (t)\right|\right|\right|}^{\frac{1}{q}}\underset{t\in \Omega }{\mathrm{supess}}\Vert X_t{\Vert }^r.$$

$${\Vert {\int }_{\Omega }{A}_t^{\ast }{X}_t{B}_t\mathrm{d}\mu (t)\Vert }_Q\le {\Vert {\int }_{\Omega }|A_t{|}^p\mathrm{d}\mu (t)\Vert }_Q^{\frac{1}{p}}{\Vert {\int }_{\Omega }|B_t{|}^q\mathrm{d}\mu (t)\Vert }_Q^{\frac{1}{q}}\underset{t\in \Omega }{\mathrm{supess}}\Vert X_t\Vert .$$

$${\Vert {\int }_{\Omega }{A}_t^{\ast }X{B}_t\mathrm{d}\mu (t)\Vert }_Q\le {\Vert {\left({\int }_{\Omega }|A_t{|}^p\mathrm{d}\mu (t)\right)}^{\frac{1}{p}}X{\left({\int }_{\Omega }|B_t{|}^q\mathrm{d}\mu (t)\right)}^{\frac{1}{q}}\Vert }_Q.$$

$$T_X(A_t)=(X_t{A}_t).$$

$$\left|\left|\left|\phi \left({\int }_{\Omega }{A}_t\mathrm{d}\mu (t)\right)\right|\right|\right|\le \left|\left|\left|{\int }_{\Omega }\phi (A_t)\mathrm{d}\mu (t)\right|\right|\right|,$$

$$A_t{\left({\int }_{\Omega }|A_t{|}^p\mathrm{d}\mu (t)+\epsilon I\right)}^{-1/p},B_t{\left({\int }_{\Omega }|B_t{|}^q\mathrm{d}\mu (t)+\epsilon I\right)}^{-1/q}\text{mbox}and$$

$${\left({\int }_{\Omega }|A_t{|}^p\mathrm{d}\mu (t)\right)}^{1/p}X{\left({\int }_{\Omega }|B_t{|}^q\mathrm{d}\mu (t)\right)}^{1/q}$$

$$\begin{gathered} {\Vert {\int }_{\Omega }{A}_t^{\ast }X{B}_t\mathrm{d}\mu (t)\Vert }_Q= \\ ={\Vert {\int }_{\Omega }{A}_t^{\ast }{\left({\int }_{\Omega }+\epsilon I\right)}^{-\frac{1}{p}}{\left({\int }_{\Omega }+\epsilon I\right)}^{\frac{1}{p}}X{\left({\int }_{\Omega }+\epsilon I\right)}^{\frac{1}{q}}{\left({\int }_{\Omega }+\epsilon I\right)}^{-\frac{1}{q}}{B}_t\mathrm{d}\mu (t)\Vert }_Q\le \\ \le {\Vert {\left({\int }_{\Omega }|A_t{|}^p\mathrm{d}\mu (t)+\epsilon I\right)}^{\frac{1}{p}}X{\left({\int }_{\Omega }|B_t{|}^q\mathrm{d}\mu (t)+\epsilon I\right)}^{\frac{1}{q}}\Vert }_Q. \end{gathered}$$

$$\left|\left|\left|{\left|\sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }{X}_n{B}_n\right|}^r\right|\right|\right|\le {\left|\left|\left|{\left(\sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }f(|X_n^{\ast }|{)}^2{A}_n\right)}^{\frac{pr}{2}}\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|{\left(\sum _{n=1}^{+\infty }{\gamma }_n{B}_n^{\ast }g(|X_n|{)}^2{B}_n\right)}^{\frac{qr}{2}}\right|\right|\right|}^{\frac{1}{q}}.$$

$$\left|\left|\left|{\left|\sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }{X}_n{B}_n\right|}^r\right|\right|\right|\le {\left|\left|\left|{\left(\sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }|X_n^{\ast }{|}^{2\alpha }{A}_n\right)}^{\frac{pr}{2}}\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|{\left(\sum _{n=1}^{+\infty }{\gamma }_n{B}_n^{\ast }|X_n{|}^{2\beta }{B}_n\right)}^{\frac{qr}{2}}\right|\right|\right|}^{\frac{1}{q}}.$$

$$\left|\left|\left|\sum _{n=1}^{+\infty }{\left|{\gamma }_n{A}_n^{\ast }{X}_n{B}_n\right|}^r\right|\right|\right|\le {\left|\left|\left|\sum _{n=1}^{+\infty }{\gamma }_n|A_n{|}^{pr}\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|\sum _{n=1}^{+\infty }{\gamma }_n|B_n{|}^{qr}\right|\right|\right|}^{\frac{1}{q}}\underset{n\ge 1}{\sup }\Vert X_n\Vert .$$

$${\Vert \sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }{X}_n{B}_n\Vert }_Q\le {\Vert \sum _{n=1}^{+\infty }{\gamma }_n|A_n{|}^p\Vert }_Q^{\frac{1}{p}}{\Vert \sum _{n=1}^{+\infty }{\gamma }_n|B_n{|}^q\Vert }_Q^{\frac{1}{q}}\underset{n\ge 1}{\sup }\Vert X_n\Vert .$$

$$\left|\left|\left|\sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }{X}_n{B}_n\right|\right|\right|\le {\left|\left|\left|\sum _{n=1}^{+\infty }{\gamma }_n|A_n{|}^p\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|\sum _{n=1}^{+\infty }{\gamma }_n^{q/2}|B_n{|}^q\right|\right|\right|}^{\frac{1}{q}}\underset{n\ge 1}{\sup }\Vert X_n\Vert .$$

$$\left|\left|\left|\sum _{n=1}^m{A}_n^{\ast }{X}_n{B}_n\right|\right|\right|\le m^{|\frac{1}{2}-\frac{1}{p}|}{\left|\left|\left|\sum _{n=1}^m|A_n{|}^p\right|\right|\right|}^{\frac{1}{p}}{\left|\left|\left|\sum _{n=1}^{+\infty }|B_n{|}^q\right|\right|\right|}^{\frac{1}{q}}\underset{n\ge 1}{\sup }\Vert X_n\Vert .$$

$${\Vert \sum _{n=1}^{+\infty }{\gamma }_n{A}_n^{\ast }X{B}_n\Vert }_Q\le {\Vert {\left(\sum _{n=1}^{+\infty }{\gamma }_n|A_n{|}^p\right)}^{\frac{1}{p}}X{\left(\sum _{n=1}^{+\infty }{\gamma }_n|B_n{|}^q\right)}^{\frac{1}{q}}\Vert }_Q.$$

$$\left|\left|\left|\sum _{n=1}^m{A}_n^{\ast }X{B}_n\right|\right|\right|\le m^{|\frac{1}{2}-\frac{1}{p}|}\left|\left|\left|{\left(\sum _{n=1}^m|A_n{|}^p\right)}^{\frac{1}{p}}X{\left(\sum _{n=1}^m|B_n{|}^q\right)}^{\frac{1}{q}}\right|\right|\right|.$$

$$\left|\left|\left|\phi (\sum B_n)\right|\right|\right|\le \left|\left|\left|\sum \phi (B_n)\right|\right|\right|$$

$$⟨x,B^q{♯}_{1/p}{A}^{p}x⟩\le ⟨x,B^{q}x⟩{♯}_{1/p}⟨x,A^{p}x⟩,$$

Full text

The author was supported in part by the Ministry of education and science, Republic of Serbia, Grant #174034.

1. Introduction

Operator inequalities have been studied for many years. Discrete version of Cauchy-Schwarz inequality can be tracked back to [15] in connections with norm estimate of elementary operators and later in [6], [7] and many others. See, also recent work [3] and references therein. Continuous version in connection with Birman-Solomyak double operator integrals was considered in [10].

In the last ten years, there was obtained many inequalities in the framework of Hilbert $C^{*}$-modules. Among others there are [5] or [8]. In [11] we established that operator Cauchy-Schwarz inequalities can be derived from the corresponding Hilbert modules inequalities in a much more easier way then previously.

Hölder type inequalities (their discrete versions) was considered in [1] and [2] (see also references therein).

The aim of this note is to use weighted Cauchy-Scwarz inequality in Hilbert modules [14, Theorem 5.1.] to obtain a very short and easy proof of many Hölder type inequalities for Hilbert space operators. They comprises known results (predominantly in discrete case) as well as completely new inequalities.

2. Preliminaries

To prove our main result we need the following:

Definition 2.1**.**

Let $A$ be a $C^{*}$-algebra and let $a$, $b\in A$ be positive elements. Its geometric mean $a\sharp b$ is defined as

[TABLE]

if $a$ is invertible and $a\sharp b=\lim_{\varepsilon\downarrow 0+}(a+\varepsilon)\sharp b$ otherwise.

It is well known that geometric mean has the following properties:

1. (1)

$a\sharp b=(ab)^{1/2}$ if $a$ and $b$ commute; 2. (2)

$a\sharp b=b\sharp a$; 3. (3)

$a\sharp b=\max\left\{0\leq x\in A\;|\;\left[\begin{matrix}a&x\\ x&b\end{matrix}\right]\geq 0\right\}$.

For more details see [4]. For the third property, see also [13].

Our main result relies on the following two theorems, first established by and the second by Horn and Mathias.

Theorem S**.**

([14, Theorem 5.1])* Let $T$ be a bounded adjointable operator on a Hilbert $C^{*}$-module $M$ that has a polar decomposition, let $x\in M$ be arbitrary, let $f$, $g:[0,+\infty)\to[0,+\infty)$ be continuous functions such that $f(t)g(t)\equiv t$. If $\left<x,Ty\right>=u|\left<x,Ty\right>|$ then*

[TABLE]

Proof.

This is proved in [14] for $f(t)=t^{\alpha}$, $g(t)=t^{\beta}$, $\alpha$, $\beta\geq 0$, $\alpha+\beta=1$, and for $T$ that has adjoint and such that the closures of the ranges of both $T$ and $T^{*}$ are complemented. However, the proof relies only on the fact that $T$ has a polar decomposition $T=U|T|$.

For the convenience of the reader, we shall give the outline of the proof. First, recall the inequality

[TABLE]

proved in [8]. Next,

[TABLE]

and apply (2.2):

[TABLE]

Finally, note that $Up(|T|)U^{*}=p(|T^{*}|)$ for all polynomials that contain only even terms, and henceforth for all continuous functions by standard limit process. ∎

Remark 2.1*.*

It will be important that $u$ can be extended either to isometry or to coisometry i.e. either $u^{*}u=1$ or $uu^{*}=1$.

Theorem HM**.**

([9, Theorem 2.3 and remark after if – formula (2.11)])* Let $A$, $B$ and $X$ be operators on a Hilbert space $H$ such that $\left[\begin{matrix}A&X\\ X^{*}&B\end{matrix}\right]\geq 0$. Then for all unitarily invariant norms $\left|\!\left|\!\left|\cdot\right|\!\right|\!\right|$, and all $p$, $q$, $r>0$ such that $1/p+1/q=1$ there holds*

[TABLE]

Remark 2.2*.*

This theorem is proved for matrices. However, carefully reading the proof we can see that everything works in the framework of Hilbert space operators, as well.

3. Unitarily invariant norm inequalities

The following general theorem leads to a large number of Hölder type inequalities.

Theorem 3.1**.**

Let $M$ be a Hilbert $C^{*}$-module over the algebra $B(H)$ of all bounded operators on some Hilbert space, let $x$, $y\in M$ and let $T:M\to M$ be adjointable operator that has a polar decomposition. If $f$, $g:[0,+\infty)\to[0,+\infty)$ are continuous functions such that $f(t)g(t)\equiv t$ then

[TABLE]

for all unitarily invariant norms $\left|\!\left|\!\left|\cdot\right|\!\right|\!\right|$ and all $p$, $q$, $r>0$ such that $1/p+1/q=1$.

Proof.

Suppose $\left<x,Ty\right>=u\left|\left<x,Ty\right>\right|$ where $u$ is extended to an coisometry, i.e. $uu^{*}=1$. Denote

[TABLE]

By Theorem S we have $\left|\left<x,Ty\right>\right|\leq u^{*}Xu\sharp Y$. Next, by property (3) of geometric mean, we have

[TABLE]

and therefore

[TABLE]

Hence by Theorem HM, formula 3.1 follows.

If $u$ can not be extended to a coisometry then we pick $\left<x,Ty\right>^{*}=\left<y,T^{*}x\right>$ which has a polar decomposition $\left<x,Ty\right>^{*}=v\left|\left<x,Ty\right>^{*}\right|$ with $v$ extendable to a coisometry. Now result follows by interchanging roles of $x$ and $y$, $T$ and $T^{*}$, $f$ and $g$. ∎

Next, we give corollaries.

Corollary 3.2**.**

Let $(\Omega,\mu)$ be a measurable space. We consider the Hilbert module $L^{2}(\Omega,\mu)$ consisting of all weakly measurable families $A_{t}\in B(H)$, $t\in\Omega$ such that $\int_{\Omega}A_{t}^{*}A_{t}\mathrm{d}\mu(t)$ exists in the weak-$*$ sense, see [11, Example 2.3.].

($i$)* If $A_{t}$, $B_{t}\in L^{2}(\Omega)$, $X_{t}\in L^{\infty}(\Omega)$, i.e. $\sup\|X_{t}\|<+\infty$ then*

[TABLE]

In particular

[TABLE]

where $\alpha$, $\beta\geq 0$, $\alpha+\beta=1$.

($ii$)* If, in addition, $\mu(\Omega)<+\infty$ and $r\geq 2$ then*

[TABLE]

The latter can be extended to all $A_{t}$, $B_{t}$ such that $\int_{\Omega}|A_{t}|^{p}\mathrm{d}\mu(t)$, $\int_{\Omega}|B_{t}|^{q}\mathrm{d}\mu(t)$ exist as weak-$8$ integrals. In particular, for any $Q$ norm $\|\cdot\|_{Q}$ we have

[TABLE]

($iii$)* If $A_{t}$ and $B_{t}$ are families consisting of mutually commuting normal operators, $X\in B(H)$ arbitrary and $\mu(\Omega)<+\infty$, then for all $Q$-norms $\|\cdot\|_{Q}$ there holds*

[TABLE]

Proof.

We have:

($i$) Consider the mapping $T_{X}:L^{2}(\Omega,\mu)\to L^{2}(\Omega,\mu)$ given by

[TABLE]

It is easy to see that $T_{X}$ is bounded, adjointable, $\|T_{x}\|=\operatorname*{sup\,ess}\|X_{t}\|$, as well as that $T_{X}$ has a polar decomposition $T_{X}=T_{U}T_{|X|}$, where $X_{t}=U_{t}|X_{t}|$ is the polar decomposition of $X_{t}$. Apply Theorem 3.1.

($ii$) First, note that $A_{t}^{*}|X^{*}_{t}|^{2\alpha}A_{t}\leq A_{t}^{*}A_{t}\|X^{*}_{t}\|^{2\alpha}\leq|A_{t}|^{2}\operatorname*{sup\,ess}\|X_{t}\|^{2\alpha}$ and similar for $B_{t}$. Then recall the Jensen inequality

[TABLE]

which holds for convex functions $\varphi:[0,+\infty)\to[0,+\infty)$, pointwise positive $A_{t}$ and $\mu(\Omega)=1$, see [12, Proposition 2.5.], and note that $t\mapsto t^{pr/2}$, $t^{qr/2}$ are convex for $r\geq 2$, $p$, $q\geq 1$. Finally, note that both sides of the required inequality are homogeneous and hence $\mu(\Omega)$ can be canceled.

For the second inequality, recall that unitarily invariant norm $\left|\!\left|\!\left|\cdot\right|\!\right|\!\right|$ is called $Q$-norm, if there is an other unitarily invariant norm $\|\cdot\|$ such that $\left|\!\left|\!\left|A\right|\!\right|\!\right|=\||A|^{2}\|^{1/2}$. Now, take $r=2$ in the previous inequality and take a power to the $1/2$.

($iii$) For $X_{t}\equiv X$ and $\int_{\Omega}|A_{t}|^{p}\mathrm{d}\mu(t)$, $\int_{\Omega}|B_{t}|^{q}\mathrm{d}\mu(t)\leq I$ the result follows from the previous item. In general case, note that $A_{t}$ as well as $A_{t}^{*}$ commutes with $\int_{\Omega}|A_{t}|^{p}\mathrm{d}\mu(t)$. Choose $\varepsilon>0$ and apply the special case to

[TABLE]

[TABLE]

instead of $A_{t}$, $B_{t}$ and $X$. We obtain

[TABLE]

Finally, let $\varepsilon\to 0$. (Notice the well known lower semicontinuity of unitarily invariant norms, with respect to weak limit.) ∎

Corollary 3.3**.**

Let us consider the discrete case. Let $\gamma_{n}\in\mathbf{R}^{+}$ be a sequence, $A_{n}$, $B_{n}$, $X_{n}\in B(H)$, and let $p$, $q$, $r>0$, $1/p+1/q=1$.

($i$)* [1, Theorem 21.] (see also Theorem 15 of the same reference) If $\sum_{n=1}^{+\infty}\gamma_{n}|A_{n}|^{2}$, $\sum_{n=1}^{+\infty}\gamma_{n}|B_{n}|^{2}$ converge weakly, $\sup\|X_{n}\|<+\infty$ and $f$, $g:[0,+\infty)\to[0,+\infty)$ are continuous functions such that $f(t)g(t)\equiv t$ then*

[TABLE]

In particular, for $\alpha$, $\beta\geq 0$, $\alpha+\beta=1$, we have

[TABLE]

($ii$)* If, in addition, $\sum_{n=1}^{+\infty}\gamma_{n}=1$ and $r\geq 2$ then*

[TABLE]

The latter can be extended to all $A_{n}$, $B_{n}$ such that $\sum|A_{n}|^{p}$, $\sum|B_{n}|^{q}$ converge weakly. In particular, for any $Q$ norm $\|\cdot\|_{Q}$ we have

[TABLE]

($iii$)* For all unitarily invariant norms $\left|\!\left|\!\left|\cdot\right|\!\right|\!\right|$ and for $p\geq 2$ there holds*

[TABLE]

In particular, for $m\in\mathbf{N}$ we have:

[TABLE]

The last inequality reduces to [1, Theorem 17] for $X_{n}\equiv I$. Note that the constant $m^{\left|\frac{1}{2}-\frac{1}{p}\right|}$ is sharp, see [1, Remark 18].

($iv$)* If $A_{n}$ and $B_{n}$ are sequences of mutually commuting normal operators, $X\in B(H)$ arbitrary and $\sum_{n=1}^{+\infty}\gamma_{n}<+\infty$ then for all $Q$-norms $\|\cdot\|_{Q}$ there holds*

[TABLE]

($v$)* If $A_{n}$ and $B_{n}$ are sequences of mutually commuting normal operators, $X\in B(H)$ arbitrary and $m\in\mathbf{N}$ then for all unitarily invariant norms $\left|\!\left|\!\left|\cdot\right|\!\right|\!\right|$ there holds*

[TABLE]

Proof.

($i$), ($ii$) and ($iv$) are special cases of ($i$), ($ii$) and ($iii$), respectively of the previous corollary, for $\Omega=\mathbf{N}$, $\mu(E)=\sum_{n\in E}\gamma_{n}$.

($iii$) Put $r=1$ and $X_{n}=I$ in (3.2). The function $t\mapsto t^{p/2}$ is convex and $t\mapsto t^{q/2}$ is concave. Apply, aforementioned Jensen inequality to the first factor, and the inequality

[TABLE]

which holds for concave $\varphi$ (see [12, Lemma 2.4]) to the second factor.

($v$) This can be derived from ($iii$) in the same way as in the previous corollary the third assertion is derived form the first.

∎

4. An inequality in operator ordering

In the paper [14] an other inequality in Hilbert modules was proved, namely

Theorem S2**.**

([14, Theorem 4.1])* Let $p>1$, $1/p+1/q=1$ and let $A$, $B$ be positive adjointable operators on some Hilbert $C^{*}$-module over a $C^{*}$-algebra. Then*

[TABLE]

where $\sharp_{\theta}$ stands for weighted geometric mean:

[TABLE]

for $A$ invertible, and $A\sharp_{\theta}B=\lim\limits_{\varepsilon\downarrow 0}(A+\varepsilon I)\sharp_{\theta}B$, otherwise.

Applying this to "diagonal" operators $X_{t}\mapsto A_{t}X_{t}$ and $X_{t}\mapsto B_{t}X_{t}$ for pointwise positive and weakly measurable families $A_{t}$ and $B_{t}$, we obtain the following corollary:

Corollary 4.1**.**

We have

1. (1)

For a measurable space $(\Omega,\mu)$ and weakly measurable families $A_{t}$, $B_{t}$, of positive Hilbert space operators, $p$, $q\geq 1$, $1/p+1/q=1$ we have

[TABLE] 2. (2)

For sequences $A_{n}$, $B_{n}$, of positive Hilbert space operators, $p$, $q\geq 1$, $1/p+1/q=1$ we have

[TABLE]