Some Gruss type inequalities for Frechet differentiable mappings

TL;DR

This paper develops generalized Gruss inequalities for Frechet differentiable mappings on Hilbert C*-modules, expanding the theoretical framework for inequalities in operator algebra contexts.

Contribution

It introduces new semi-inner products on differentiable function spaces and establishes Gruss type inequalities within this generalized setting.

Findings

Established Gruss inequalities in semi-inner product C*-modules.

Introduced generalized semi-inner products for differentiable mappings.

Extended inequalities to mappings on Hilbert C*-modules.

Abstract

Let X be a Hilbert C^*-module on C^*-algebra A and p in A. We denote by Dp(A;X) the set of all continuous functions f on A, which are Frechet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A;X), and using them some Gruss type inequalities in semi-inner product C^*-module Dp(A;X) and Dp(A;X^n) are established.

Equations

$$\underset{h\to 0}{\lim }\frac{\Vert f(p+h)-f(p)-u(h){\Vert }_X}{\Vert h{\Vert }_A}=0,$$

$$\Vert f(x)-f(y)\le \Vert x-y\Vert \underset{0<\theta <1}{\sup }\Vert Df((1-\theta )x+\theta y)\Vert .$$

$$T(f,g):=\frac{1}{b-a}{\int }_a^{b}f(t)g(t)dt-\frac{1}{b-a}{\int }_a^{b}f(t)dt\frac{1}{b-a}{\int }_a^{b}g(t)dt.$$

$$\left|T(f,g)\right|\le \frac{1}{4}(M-m)(N-n),$$

$$f(x)=g(x)=sgn\Big{(}x-\frac{a+b}{2}\Big{)}.$$

$$\left|\frac{1}{n}\sum _{i=1}^n{a}_i{b}_i-\frac{1}{n}\sum _{i=1}^n{a}_i.\frac{1}{n}\sum _{i=1}^n{b}_i\right|\le \frac{1}{4}(A-a)(B-b),$$

$$Re⟨X-x_i,x_i-x⟩\ge 0foralli\in \{1,...,n\},$$

$$\Vert x_i-\frac{x+X}{2}\Vert \le \frac{1}{2}\Vert X-x\Vert foralli\in \{1,...,n\},$$

$$\Vert \sum _{i=1}^n{p}_i{\alpha }_i{x}_i-\sum _{i=1}^n{p}_i{\alpha }_i\sum _{i=1}^n{p}_i{x}_i\Vert$$

$$\le \frac{1}{2}\Vert X-x\Vert {\left[\sum _{i=1}^n{p}_i|{\alpha }_i{|}^2-{\left|\sum _{i=1}^n{p}_i{\alpha }_i\right|}^2\right]}^{\frac{1}{2}}.$$

$$⟨x,y⟩⟨y,x⟩\le \Vert ⟨x,x⟩\Vert ⟨y,y⟩(x,y\in X)$$

$$\Vert x-\frac{\alpha +\beta }{2}e\Vert \le \frac{1}{2}|\alpha -\beta |,\Vert y-\frac{\lambda +\mu }{2}e\Vert \le \frac{1}{2}|\lambda -\mu |,$$

$$\Vert ⟨x,y⟩-⟨x,e⟩⟨e,y⟩\Vert \le \frac{1}{4}\mid \alpha -\beta ||\lambda -\mu \mid .$$

$$Dk(p)(a)=⟨Df(p)(a),g(p)⟩+⟨f(p),Dg(p)(a)⟩.$$

$$⟨f(p+h),g(p+h)⟩-⟨f(p),g(p)⟩-⟨Df(p)(h),g(p)⟩-⟨f(p),Dg(p)(h)⟩$$

$$=⟨f(p+h),g(p+h)-g(p)-Dg(p)(h))⟩+⟨f(p+h)-f(p),Dg(p)(h)⟩$$

$$+⟨f(p+h)-f(p)-Df(p)(h),g(p)⟩.$$

$$[f,g]_{a}:=\big{\langle}Df(p)(a),Dg(p)(a)\big{\rangle}_{1}+\big{\langle}f(p),g(p)\big{\rangle}_{1}-D\big{\langle}f(\cdot),g(\cdot)\big{\rangle}_{1}(p)(a),$$

$$⟨f(a),g(a){⟩}_1=⟨f(a),g(a)⟩-⟨f(a),e⟩⟨e,g(a)⟩.$$

$$\Vert f_a(p+h)-f_a(p)-(Df(p)(h))a\Vert$$

$$\le \Vert f(p+h)-f(p)-Df(p)(h)\Vert \Vert a\Vert .$$

$$[f,g]_{a}=\Big{\langle}Df(p)(a)-f(p),Dg(p)(a)-g(p)\rangle\\ -\langle Df(p)(a)-f(p),e\rangle\langle e,Dg(p)(a)-g(p)\Big{\rangle}\\ =\Big{\langle}(Df(p)(a)-f(p))-e\big{\langle}e,(Df(p)(a)-f(p))\big{\rangle}\\ ,(Dg(p)(a)-g(p))-e\big{\langle}e,(Dg(p)(a)-g(p))\big{\rangle}\Big{\rangle}.$$

$$[f,f]_{a}=\Big{\langle}(Df(p)(a)-f(p))-e\big{\langle}e,(Df(p)(a)-f(p))\big{\rangle}\\ ,(Df(p)(a)-f(p))-e\big{\langle}e,(Df(p)(a)-f(p))\big{\rangle}\Big{\rangle}\geq 0.$$

$$\Vert f(p)-\frac{\alpha +\beta }{2}e\Vert \le \frac{1}{2}|\alpha -\beta |$$

$$\Vert Df(p)(a)-\frac{{\alpha }^{\prime }+{\beta }^{\prime }}{2}e\Vert \le \frac{1}{2}|{\alpha }^{\prime }-{\beta }^{\prime }|$$

$$\Vert g(p)-\frac{\lambda +\mu }{2}e\Vert \le \frac{1}{2}|\lambda -\mu |$$

$$\Vert Dg(p)(a)-\frac{{\mu }^{\prime }+{\lambda }^{\prime }}{2}e\Vert \le \frac{1}{2}|{\mu }^{\prime }-{\lambda }^{\prime }|,$$

$$\|\langle Df(p)(a),Dg(p)(a)\rangle_{1}+\langle f(p),g(p)\rangle_{1}-D\big{\langle}f(\cdot),g(\cdot)\big{\rangle}_{1}(p)(a)\|\\ \leq\frac{1}{2}(|\alpha-\beta|+|\alpha^{\prime}-\beta^{\prime}|)(|\lambda-\mu|+|\lambda^{\prime}-\mu^{\prime}|).\\ \\$$

$$\Vert [f,g{]}_a{\Vert }^2\le \Vert [f,f{]}_a\Vert \Vert [g,g{]}_a\Vert .$$

$$\|[f,f]_{a}\|\leq\big{\|}\langle Df(p)(a),Df(p)(a)\rangle-\langle Df(p)(a),e\rangle\langle e,Df(p)(a)\rangle\big{\|}\\ +\big{\|}\langle f(p),f(p)\rangle-\langle f(p),e\rangle\langle e,f(p)\rangle\big{\|}\\ +\big{\|}\langle Df(p)(a),f(p)\rangle-\langle Df(p)(a),e\rangle\langle e,f(p)\rangle\big{\|}\\ +\big{\|}\langle f(p),Df(p)(a)\rangle-\langle f(p),e\rangle\langle e,Df(p)(a)\rangle\big{\|}.$$

$$\begin{gathered} \Vert [f,f{]}_a\Vert \le \frac{1}{4}|{\alpha }^{\prime }-{\beta }^{\prime }{|}^2+\frac{1}{4}|\alpha -\beta {|}^2+\frac{1}{2}|{\alpha }^{\prime }-{\beta }^{\prime }||\alpha -\beta | \\ =\frac{1}{4}(|\alpha -\beta |+|{\alpha }^{\prime }-{\beta }^{\prime }|{)}^2. \end{gathered}$$

$$\begin{gathered} \Vert [g,g{]}_a\Vert \le \frac{1}{4}|{\lambda }^{\prime }-{\mu }^{\prime }{|}^2+\frac{1}{4}|\lambda -\mu {|}^2+\frac{1}{2}|{\lambda }^{\prime }-{\mu }^{\prime }||\lambda -\mu | \\ =\frac{1}{4}(|\lambda -\mu |+|{\lambda }^{\prime }-{\mu }^{\prime }|{)}^2. \end{gathered}$$

$$\Vert Df(p)-\widehat{\frac{x_0+y_0}{2}}\Vert \le \Vert \frac{x_0-y_0}{2}\Vert ,$$

$$\Vert Dg(p)-\widehat{\frac{x_1+y_1}{2}}\Vert \le \Vert \frac{x_1-y_1}{2}\Vert ,$$

Full text

*Corresponding author

1. Introduction

Let $A,X$ be two normed vector spaces over $\mathbb{K}(\mathbb{K}=\mathbb{C},\mathbb{R})$, we recall that a function $f:A\rightarrow X$ is Fréchet differentiable in $p\in A$, if there exists a bounded linear mapping $u:A\rightarrow X$ such that

[TABLE]

and in this case, we denote $u$ by $Df(p)$. Let $D_{p}(A,X)$ denotes the set of all continuous functions $f:A\rightarrow X$, which are Fréchet differentiable on a open neighborhood (say $U$) of $p$. The main purpose of differential calculus consists in getting some information using an affine approximation to a given nonlinear map around a given point. In many applications it is important to have Fréchet derivatives of $f$, since they provide genuine local linear approximation to $f$. For instance let $U$ be an open subset of $A$ containing the segment $[x,y]=\{(1-\theta)x+\theta y:0\leq\theta\leq 1\}$, and let $f:A\rightarrow X$ be Fréchet differentiable on $U$, then the following mean value formula holds

[TABLE]

For two Lebesgue integrable functions $f,g:[a,b]\rightarrow\mathbb{R}$, consider the Čeby$\breve{s}$ev functional:

[TABLE]

In 1934, G. Grüss [4] showed that

[TABLE]

provided $m,M,n,N$ are real numbers with the property $-\infty<m\leq f\leq M<\infty$ and $-\infty<n\leq g\leq N<\infty\quad\text{a.e. on }[a,b].$ The constant $\frac{1}{4}$ is best possible in the sense that it cannot be replaced by a smaller quantity and is achieved for

[TABLE]

The discrete version of (1.3) states that: If $a\leq a_{i}\leq A,~{}b\leq b_{i}\leq B,~{}(i=1,...,n)$ where $a,A,b,B,a_{i},b_{i}$ are real numbers, then

[TABLE]

where the constant $\frac{1}{4}$ is the best possible for an arbitrary $n\geq 1$. Some refinements of the discrete version of Grüss inequality (1.4) for inner product spaces are available in [1, 6].

Theorem 1**.**

Let $(H;\langle\cdot,\cdot\rangle)$ and $\mathbb{K}$ be as above and $\overline{x}=(x_{1},...,x_{n})\in H^{n}$, $\overline{\alpha}=(\alpha_{1},...,\alpha_{n})\in\mathbb{K}^{n}$ and $\overline{p}=(p_{1},...,p_{n})$ a probability vector. If $x,X\in H$ are such that

[TABLE]

or, equivalently,

[TABLE]

holds, then the following inequality holds

[TABLE]

The constant $\frac{1}{2}$ in the first and second inequalities is best possible.

In recent years several refinements and generalizations have been considered for the Grüss inequality. We would like to refer the reader to [2, 3, 4, 5, 6, 8, 9] and references therein for more information.

In this paper, for every Hilbert $C^{*}$-module $X$ over a $C^{*}$-algebra $A$, some Grüss type inequalities in semi-inner product $C^{*}$-module $D_{p}(A,X^{n})$ are established. We also for two arbitrary Banach $*$-algebras, define a norm and an involution map on $D_{p}(A,B)$ and prove that $D_{p}(A,B)$ is a Banach $*$-algebra.

2. Grüss type inequalities for differentiable mappings

Let $A$ be a $C^{*}$-algebra. A semi-inner product module over $A$ is a right module $X$ over $A$ together with a generalized semi-inner product, that is with a mapping $\langle.,.\rangle$ on $X\times X$, which is $A$-valued and has the following properties:

1. (i)

$\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$ for all $x,y,z\in X,$ 2. (ii)

$\left\langle x,ya\right\rangle=\left\langle x,y\right\rangle a$ for $x,y\in X,a\in A$, 3. (iii)

$\langle x,y\rangle^{*}=\langle y,x\rangle$ for all $x,y\in X$, 4. (iv)

$\left\langle x,x\right\rangle\geq 0$ for $x\in X$.

We will say that $X$ is a semi-inner product $C^{*}$-module. If, in addition,

1. (v)

$\langle x,x\rangle=0$ implies $x=0$,

then $\langle.,.\rangle$ is called a generalized inner product and $X$ is called an inner product module over $A$ or an inner product $C^{*}$-module. An inner product $C^{*}$-module which is complete with respect to its norm $\|x\|=\|\langle x,x\rangle\|^{\frac{1}{2}}$, is called a Hilbert $C^{*}$-module.

As we can see, an inner product module obeys the same axioms as an ordinary inner product space, except that the inner product takes values in a more general structure rather than in the field of complex numbers.

If $A$ is a $C^{*}$-algebra and $X$ is a semi-inner product $A$-module, then the following Schwarz inequality holds:

[TABLE]

($e.g.$ [7, Proposition 1.1]).

Theorem 2**.**

[3]** Let $A$ be a $C^{*}$- Algebra, $X$ a Hilbert $C^{*}$- module. If $x,y,e\in X$, $\langle e,e\rangle$ is an idempotent in $A$ and $\alpha,\beta,\lambda,\mu$ are complex numbers such that

[TABLE]

hold, then one has the following inequality;

[TABLE]

Example 1**.**

Let $A$ be a real $C^{*}$-algebra and $X$ be a semi-inner product $C^{*}$-module on a $C^{*}$-algebra $B$. If functions $f,g\in D_{p}(A,X)$, then function $k:A\rightarrow B$ as $k(a)=\langle f(a),g(a)\rangle$ is differentiable in $(p\in A)$ and derivative of that is a linear mapping $Dk(p):A\rightarrow B$ defined by

[TABLE]

Because

[TABLE]

Let $A$ be a $C^{*}$-algebra and $X$ a semi-inner product $A$-module. If $f\in D_{p}(A,X)$ and $a\in A$, we define the function $f_{a}:A\rightarrow X$ by $f_{a}(t)=f(t)a$.

Theorem 3**.**

Let $X$ be a semi-inner product $C^{*}$-module on $C^{*}$-algebra $A$, and $p\in A,e\in X$. If $\langle e,e\rangle$ is an idempotent element in $A$, and $f,g\in D_{p}(A,X)$, then for every $a\in A$, the map $[\cdot,\cdot]_{a}:D_{p}(A,X)\times D_{p}(A,X)\rightarrow A$ with;

[TABLE]

is a generalized semi-inner product on $D_{p}(A,X)$, where

[TABLE]

Proof.

First, we show that $f_{a}\in D_{p}(A,X)$ and $Df_{a}(p)=(Df(p))a$. There exists a bounded convex set $V(=B(p,r))$ containing $p$ such that $V\subseteq U$. Let $p,h\in V,a\in A$, then

[TABLE]

This implies that $f_{a}\in D_{p}(A,X)$.

A simple calculation shows

[TABLE]

Therefore,

[TABLE]

It is easy to show that $[\cdot,\cdot]_{a}$ is a generalized semi-inner product on $D_{p}(A,X)$. ∎

Lemma 1**.**

*Let $X$ be a semi-inner product $C^{*}$-module on $C^{*}$-algebra $A$, and $p,a\in A,e\in X$. If $\langle e,e\rangle$ is an idempotent element in $A$, $f,g\in D_{p}(A,X)$, and

$\alpha,\beta,\alpha^{\prime},\beta^{\prime},\mu,\lambda,\mu^{\prime},\lambda^{\prime}$ are complex numbers such that*

[TABLE]

then the following inequality holds

[TABLE]

Proof.

Since $[\cdot,\cdot]_{a}$ is a generalized semi-inner product on $D_{p}(A,X)$, the Schwartz inequality holds, i.e,

[TABLE]

We know that

[TABLE]

This inequality and Theorem 2 imply that

[TABLE]

Similarly

[TABLE]

∎

Let $X$ be a semi-inner product $C^{\ast}$-module over $C^{*}$-algebra $A$. For every $x\in X$, we define the map $\hat{x}:A\rightarrow X^{n}$ by $\hat{x}(a)=(xa,...,xa),~{}~{}~{}~{}(a\in A)$.

Lemma 2**.**

Let $X$ be a semi-inner product $C^{\ast}$-module, $x_{0},y_{0},x_{1},y_{1}\in X$ and $(r_{1},r_{2},....,r_{n})\in\mathbb{R}^{n}$ a probability vector. If $p\in A$ and $f=(f_{1},...,f_{n}),g=(g_{1},...,g_{n})\in D_{p}(A,X^{n})$ such that

[TABLE]

and

[TABLE]

then for all $a\in A$, we have

[TABLE]

Proof.

For every $a\in A$, we define the map \big{(}\cdot,\cdot\big{)}_{a}:D_{p}(A,X^{n})\times D_{p}(A,X^{n})\rightarrow A with;

[TABLE]

The following Korkine type inequality for differentiable mappings holds:

[TABLE]

Therefore, \big{(}f,f\big{)}_{a}\geq 0. It is easy to show that \big{(}\cdot,\cdot\big{)}_{a} is a generalized semi-inner product on $D_{p}(A,X^{n})$.

A simple calculation shows that

[TABLE]

From Schwartz inequality, we have

[TABLE]

∎

Corollary 1**.**

Let $X$ be a semi-inner product $C^{\ast}$-module, $x_{0},y_{0}\in X,(\alpha_{1},...,\alpha_{n})\in\mathbb{C}^{n}$ and $(r_{1},r_{2},....,r_{n})\in\mathbb{R}^{n}$ a probability vector. If $p\in A$ and $f=(f_{1},...,f_{n})\in D_{p}(A,X^{n})$ such that

[TABLE]

then for all $a\in A$, we have

[TABLE]

Proof.

[TABLE]

∎

Corollary 2**.**

Let $X$ be a semi-inner product $C^{*}$-module, $x_{0},y_{0}\in X$. If $p\in A$ and $f=(f_{1},...,f_{n})\in D_{p}(A,X^{n})$ such that

[TABLE]

then for all $a\in A$, we have

[TABLE]

and

[TABLE]

Proof.

If we put $ri=\frac{1}{n},\alpha_{i}=k$ in inequality (1), then we get (2.4), and if $ri=\frac{1}{n},\alpha_{i}=k^{2}$ in inequality (1), then we get (2.5). ∎

3. Differential mapping on Banach *-algebras

Theorem 4**.**

Let $A,B$ be two Banach $*$-algebras and $p$ be a self adjoint element in $A$. Then $D_{p}(A,B)$ is a Banach $*$-algebra with the point-wise operations and the involution $f^{*}(a)=(f(a^{*}))^{*}~{}~{}(a\in A)$, and the norm

[TABLE]

Proof.

First we show that the involution $f\rightarrow f^{*}$ is differentiable and $Df^{*}(x)(h)=(Df(x^{*})(h^{*}))^{*}~{}~{}~{}~{}~{}(x,h\in U=B(p,r))$. It is trivial that if $x\in U$, then $x^{*}\in U$ because $\|x-p\|=\|(x-p)^{*}\|=\|x^{*}-p\|$. It can be shown easily that $Df^{*}(x)$ is a bounded linear map with $\|Df^{*}(x)\|=\|Df(x^{*})\|$. Therefore

[TABLE]

From $\|Df^{*}(x)\|=\|Df(x^{*})\|$ and $\|f^{*}(a)\|=\|f(a^{*})\|$, we obtain

[TABLE]

Now, we show that $D_{p}(A,B)$ is complete. There exists a bounded convex set $V(=B(p,r))$ containing $p$ such that $V\subseteq U$. Suppose that $(f_{n})$ is a Cauchy sequence in $D_{p}(A,B)$, i.e.,

[TABLE]

Since $B$ is complete, therefore $L(A,B)$ the space of all bounded linear maps from $A$ into $B$, is complete. So there are functions $f,g$ such that $\sup_{a\in A}\|f_{n}(a)-f(a)\|\rightarrow 0$ and $\sup_{x\in V}\|Df_{n}(x)-g(x)\|\rightarrow 0$. Given $\varepsilon>0$, we can find $N\in\mathbb{N}$ such that for $m>n\geq N$ one has

[TABLE]

We may suppose that there exist $a\in A$ such that, $p+a\in V$. Using Lipschitzian functions $f_{m}-f_{n}$, we obtain that

[TABLE]

passing to the limit on $m$, we get

[TABLE]

Utilizing differentiability $f_{N}$ and (3.3), we have

[TABLE]

From (3.4) and (3), we obtain

[TABLE]

Therefore $D_{p}(A,B)$ is a Banach $*$-algebra. ∎