In this paper, we give characterizations of certain properties of inner product type integral transformers. We first consider unitarily invariant norms and operator valued functions. We then give results on norm inequalities for inner product type integral transformers in terms of Landau inequality, Grüss inequality. Lastly, we explore some of the applications in quantum theory.
Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{B(H)}\) be the algebra of all bounded linear operators on \(\mathcal{H}.\) In this paper, we discuss various types of norm inequalities for inner product type integral transformers in terms of Landau type inequality, Grüss type inequality and Cauchy-Schwarz type inequality. We shall also consider the applications in quantum theory. We begin by the following definition:
Definition 1. Grüss inequality states that if \(f\) and \(g\) are integrable real functions on \([a,b]\) such that \(C\leq f(x)\le D\) and \(E\leq g(x)\le F\) hold for some real constants \(C,D,E,F\) and for all \(x\in[a,b]\), then
Next, we discuss a very important definition of inner product type integral \((i.p.t.i)\) transformer which is key to our study.
Definition 2. Consider weakly \(\mu^*\)-measurable operator valued \((o.v)\) functions \(A, B:\Omega\rightarrow \mathcal{B(H)}\) and for all \(X\in \mathcal{B(H)}\). Let the function \(t\rightarrow A_t X B_t\) be also weakly \(\mu^*\)-measurable. If these functions are Gel’fand integrable for all \(X\in \mathcal{B(H)}\), then the inner product type linear transformation \(X\to\int_\Omega A_t X B_t dt\) is called an inner product type integral \((i.p.t.i)\) transformer on \(\mathcal{B(H)}\) and denoted by \(\int_\Omega A_t \otimes B_t dt\) or \({\mathcal I}_{A,B}\).
Remark 1. If \(\mu\) is the counting measure on \(\mathbb N\) then such transformers are known as elementary operators whose certain properties have been studied in details (see [3] and the references therein).
The set \(\mathcal{C}_{|||\cdot|||}=\{A \in \mathcal{K}(\mathcal{H}) : \left\vert \left\vert \left\vert A \right\vert \right\vert \right\vert < \infty \}\) is a closed self-adjoint ideal \(\mathcal{J}\) of \(\mathcal{B}( \mathcal{H})\) containing finite rank operators. It enjoys the following properties. First, for all \(A,B\in \mathcal{B(H)}\) and \(X \in \mathcal{J}\), \( \left\vert \left\vert \left\vert AXB\right\vert \right\vert \right\vert \leq \left\vert \left\vert A\right\vert \right\vert \ \left\vert \left\vert \left\vert X\right\vert \right\vert \right\vert \ \left\vert \left\vert B\right\vert \right\vert\,. \) Secondly, if \(X\) is a rank one operator, then \( \left\vert \left\vert \left\vert X\right\vert \right\vert \right\vert =\|X\|\,. \) The Ky Fan norm as an example of unitarily invariant norms is defined by \(\| A\| _{(k)}=\sum_{j=1}^{k}s_{j}(A)\) for \(k=1,2,\ldots\). The Ky Fan dominance Theorem [5] states that \(\| A\| _{(k)}\leq \| B\| _{(k)}\,\,(k=1,2,\ldots )\) if and only if \(|||A||| \leq |||B|||\) for all unitarily invariant norms \(|||\cdot|||\), see [6] for more information on unitarily invariant norms. The inequalities involving unitarily invariant norms have been of special interest (see [5] and the references therein).
Lemma 1. Let \(\mathcal{T}\) and \(\mathcal{S}\) be linear mappings defined on \(\mathcal{C}_\infty(\mathcal{H}).\) If \(\|\mathcal{T}X\|\le\|\mathcal{S}X\|\mbox{ for all }X\in \mathcal{C}_\infty(\mathcal{H}), \;\|\mathcal{T}X\|_1\le\|\mathcal{S}X\|_1\mbox{ for all }X\in \mathcal{C}_\infty(\mathcal{H})\), then \( \mathcal{T}X\le\mathcal{S}X\) for all unitarily invariant norms.
Proof. The norms \(\|\cdot\|\) and \(\|\cdot\|_1\) are dual to each other in the sense that \(\|X\|=\sup_{\|Y\|_1=1}|tr(XY)|\) and \( \|X\|_1=\sup_{\|Y\|=1}|tr(XY)|.\) Hence \(\|\mathcal{T}^*X\|\le\|\mathcal{S}^*X\|\) and \(\|\mathcal{T}^*X\|_1\le\|\mathcal{S}^*X\|_1\). Consider the Ky Fan norm \(\|\cdot\|_{(k)}\). Its dual norm is \(\|\cdot\|_{(k)}^\sharp=\max\{\|\cdot\|,(1/k)\|\cdot\|_1\}\). Thus, by duality, \(\|\mathcal{T}X\|_{(k)}\le\|\mathcal{S}X\|_{(k)}\) and the result follows by Ky Fan dominance property [6].
An operator \(A\in \mathcal{B(H)}\) is called \(G_{1}\) operator if the growth condition
\[ \left\Vert (z-A)^{-1}\right\Vert =\frac{1}{{dist}(z,\sigma (A))} \] holds for all \(z\) not in the spectrum \(\sigma (A)\) of \(A\). Here \({dist}(z,\sigma (A))\) denotes the distance between \(z\) and \(\sigma (A)\). It is known that hyponormal (in particular, normal) operators are \( G_{1}\) operators [4].Let \(A, B\in \mathcal{B(H)}\) and let \(f\) be a function which is analytic on an open neighborhood \( \Omega \) of \(\sigma (A)\) in the complex plane. Then \(f(A)\) denotes the operator defined on \(\mathcal{H}\) by \( f(A)=\frac{1}{2\pi i}\int\limits_{C}f(z)(z-A)^{-1}dz, \label{4} \) called the Riesz-Dunford integral, where \(C\) is a positively oriented simple closed rectifiable contour surrounding \(\sigma (A)\) in \(\Omega \) (see [2] and the references therein). The spectral mapping theorem asserts that \(\sigma (f(A))=f(\sigma (A))\). Throughout this paper, \(\mathbb{D}=\{z\in\mathbb{C}:\left\vert z\right\vert 0 \mbox{   and  } f(0)=1\}.\)
In this work, we present some upper bounds for \(|||f(A)Xg(B)\pm X|||\), where \(A, B\) are \(G_{1}\) operators, \(|||\cdot|||\) is a unitarily invariant norm and \(f, g\in \mathfrak{H}\). Further, we find some new upper bounds for the the Schatten \(2\)-norm of \(f(A)X\pm Xg(B)\). Up-to this juncture, we find some upper estimates for \(|||f(A)Xg(B)+ X|||\) in terms of \(|||\,|AXB|+|X|\,|||\) and \(|||f(A)Xg(B)- X|||\) in terms of \(|||\,|AX|+|XB|\,|||\), where \(A, B\) are \(G_{1}\) operators and \(f, g\in \mathcal{H}\).
Proposition 1. If \(A,B\in \mathcal{B(H)}\) are \(G_{1}\) operators with \(\sigma (A)\cup \sigma (B)\subset\mathbb{D}\) and \(f, g \in \mathcal{H}\), then for every \(X\in \mathcal{B(H)}\) and for every unitarily invariant norm \(\left\vert \left\vert \left\vert\cdot \right\vert \right\vert \right\vert \), the inequality \( \left\vert \left\vert \left\vert f(A)Xg(B)+X\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}d_{B}} \left\vert\left\vert \left\vert\,|AXB|+|X|\,\right\vert \right\vert \right\vert \label{5} \) holds.
Proof. From the Herglotz representation Theorem [1], it follows that \(f\in \mathcal{H}\) can be represented as
Theorem 1. Let \(f, g\in \mathcal{H}\) and \(A\in\mathcal{B(H)}\) be a \(G_{1}\) operator with \(\sigma (A)\subset\mathbb{D}\). The inequality \( \left\vert \left\vert \left\vert f(A)Xg(A^*)+X\right\vert \right\vert \right\vert \leq \frac{2}{d_{A}^2} \left\vert\left\vert \left\vert\ A|X|A^*+|X|\ \right\vert \right\vert \right\vert \) holds for every normal operator \(X\in\mathcal{B(H)}\) commuting with \(A\) and for every unitarily invariant norm \(\left\vert \left\vert \left\vert\cdot \right\vert \right\vert \right\vert \).
Proof. Let \(X\) and \(AXB\) be normal. Since \(||| C+D |||\leq |||\,|C|+|D|\,|||\) for any normal operators \(C\) and \(D\), the constant \(\sqrt{2}\) can be reduced to \(1\) in Equation (4). Now from Fuglede-Putnam theorem, if \(A\in \mathcal{B(H)}\) is an operator, \(X\in {\mathcal(B)}({\mathcal(H)})\) is normal and \(AX=XA\), then \(AX^*=X^*A\). Thus if \(X\) is a normal operator commuting with a \(G_{1}\) operator \(A\), then \(AXA^*\) is normal, \(|AXA^*|=A|X|A^*\) and \(A^*\) is a \(G_1\) operator with \(d_{A^*}=d_A\). By Proposition 1 the proof is complete.
Next, letting \(A=B\) in Proposition 1, we obtain the following result.
Corollary 1. Let \(f, g\in \mathcal{H}\) and \(A\in \mathcal{B(H)}\) be a \(G_{1}\) operator with \(\sigma (A)\subset\mathbb{D}\). Then \( \left\vert \left\vert \left\vert f(A)Xg(A)-X\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}^2} \left\vert\left\vert \left\vert \,|AX|+|XA|\,\right\vert \right\vert \right\vert \) for every \(X\in\mathcal{B}(\mathcal{H})\) and for every unitarily invariant norm \(\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert \).
Setting \(X=I\) in Proposition 1 again, we obtain the following result.
Corollary 2. Let \(f, g\in \mathfrak{H}\) and \(A,B\in\mathbb{M}_n\) be \(G_{1}\) matrices such that \(\sigma (A)\cup \sigma (B)\subset\mathbb{D}\). Then \( \left\vert \left\vert \left\vert f(A)g(B)+I\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}d_{B}} \left\vert\left\vert \left\vert\,|AB|+I\,\right\vert \right\vert \right\vert \) for every unitarily invariant norm \(\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert. \)
Corollary 3. If \(A\in \mathcal{B}(\mathcal{H})\) is self-adjoint and \(f\) is a continuous complex function on \(\sigma(A)\), then \(f(UAU^*)=Uf(A)U^*\) for all unitaries \(U\).
Proof. By the Stone-Weierstrass theorem, there is a sequence \((p_n)\) of polynomials uniformly converging to \(f\) on \(\sigma(A)\). Hence, \[f(UAU^*)=\lim_np_n(UAU^*)=U(\lim_np_n(A))U^*=Uf(A)U^*.\] We note that \(\sigma(UAU^*)=\sigma(A)\).
We conclude this section by presenting some inequalities involving the Hilbert-Schmidt norm \(\|\cdot\|_2.\)
Theorem 2. Let \(A,B\in\mathbb{M}_n\) be Hermitian matrices satisfying \(\sigma(A)\cup \sigma(B)\subset \mathbb{D}\) and let \(f, g\in \mathfrak{H}\). Then \( \|f(A)X\pm Xg(B)\|_2\leq \left\|\frac{X+|A|X}{d_A}+\frac{X+X|B|}{d_B}\right\|_2. \)
Proof. Let \(A=UD(\nu_j)U^*\) and \(B=VD(\mu_k)V^*\) be the spectral decomposition of \(A\) and \(B\) and let \(Y=U^*XV:=[y_{jk}].\) Noting that \(|e^{i\alpha}-\lambda_j|\geq d_A\) and \(|e^{i\beta}-\mu_k|\geq d_B,\) we have from [7] that \begin{align*} \|f(A)X\pm Xg(B)\|_2^2&=\sum_{j,k}|f(\lambda_j)\pm g(\mu_k)|^2|y_{jk}|^2\\&\leq\sum_{j,k}\left(\frac{1+|\lambda_j|}{d_A}+\frac{1+|\mu_k|}{d_B}\right)^2|y_{jk}|^2\\ &=\left\|\frac{X+|A|X}{d_A}+\frac{X+X|B|}{d_B}\right\|_2^2, \end{align*} which completes the proof.
Proposition 2. \(A:\Omega \rightarrow \mathcal{B(H)}\) is \([\mu]\) if and only if scalar valued functions \(t \rightarrow \langle A_{t} f,f\rangle\) are \([\mu]\) measurable (resp. integrable) for every \(f\in \mathcal{H}\).
Proof. Every one dimensional operator \(f^{*} \otimes f\) is in \(C_{1}(H)\) and \(tr(A_{t}( f^{*} \otimes f))=tr(f^{*} \otimes A_{t} f)=\left,\) so that \([\mu]\) weak \(^*\)-measurability (resp. \([\mu]\) weak \(^*\)-integrability) of \(A\) directly implies measurability (resp. integrability) of \(\left\) for any \(f\in \mathcal{H}\). The converse follows immediately from [4] and this completes the proof.
We note that in view of Proposition 2, the Equation (5) of Gel’fand integral for \(o.v.\) functions can be reformulated as follows [2]:
Proposition 3.
If \(\left\in L^1(E,\mu)\) for all \(f\in \mathcal{H}\), for
some \(E\in \mathcal{M}\) and a \(\mathcal{B(H)}\)-valued function \(A\) on \(E\), then
the mapping \(f\rightarrow\int_E \leftd\mu(t)\) represents a
quadratic form of bounded operator
\(\int_E A dm\) or \(\int_E A_t d\mu(t)\),
satisfying the following
\(
\left=
\int_E \left\,d\mu (t), for\;\; all\;\; f,g\in \mathcal{H}.\)
Proof.
It suffices to show that for all \(E\in \mathcal{M},\)
\(
\Phi_E(f,g)=\int_E \left\,d\mu (t),\)
for all \(f, g\in \mathcal{H}\),
defines a bounded sesquilinear functional \(\Phi\) on \(\mathcal{H}\).
Indeed, by [1], we have
\(
| \Phi_E(f,g)| \le \int_E|\left|\,d\mu
(t)
\le \| A_{t} f,g\|_{L^1}
\le M \|f\|\|g\|
\)
for all \(f,g\in \mathcal{H}\) since integration is a contractive functional on
\(L^{1}(\Omega ,\mu)\). This completes the proof.
Remark 2.
It is known from [1] that
for a \([\mu]\)
\(A:\Omega \rightarrow \mathcal{B(H)}\) we have that \(A^*A\) is Gel’fand
integrable if and only if \( \int_\Omega \|A_t f\|^2d\mu
(t)< \infty,\) for all
\(f\in \mathcal{H}\). Moreover,
for a \([\mu]\)
function
\(A:\Omega \rightarrow \mathcal{B(H)}\). Let us consider a linear transformation \(\vec{A}:D_{\vec{A}}\rightarrow L^{2}(\Omega,\mu , \mathcal{H})\),
with the domain \(D_{\vec{A}}=\{ f\in \mathcal{H} \,| \,
\int_\Omega \|A_t f\|^2 d\mu
(t)< \infty\}\), defined by
\(
({\vec{A}}f)(t)=A_t f .\) and all \(f\in
D_{\vec{A}}.\)
In the next section, we devote our efforts to results on inner product
type integral transformers in terms of Landau, Cauchy-Schwarz
and Grüss type norm inequalities.4. Norm inequalities
In this section, we consider various types of norm inequalities for inner product
type integral transformers discussed in [1,2,4,7].
From [1], a sufficient condition is
provided when \(A^*\) and \(B\) from Definition 2 are both in
\(L^2_G(\Omega,d\mu, \mathcal{B(H)}).\) If each of families \((A_t)_{t\in\Omega}\)
and \((B_t)_{t\in\Omega}\) consists of commuting normal operators,
then by Theorem 3.2 [1], the \(i.p.t.i\) transformer \(\int_\Omega
A_t \otimes B_t d\mu(t)\) leaves every \(u.i.\) norm ideal
\(\mathcal{C}_{|\|\cdot|\|}(\mathcal{H})\) invariant and the following
Cauchy-Schwarz inequality holds:
Proposition 4. Let \(\mu\) be a probability measure on \(\Omega\), then for every field \((\mathcal{A}_t)_{t\in\Omega}\) in \(L^2(\Omega,\mu,\mathcal{B}(\mathcal{H}))\), for all \(B\in\mathcal{B}(\mathcal{H})\), for all unitarily invariant norms \(|\|\cdot|\|\) and for all \(\theta>0\),
Proof. The expression (9) is trivial and the inequality (10) follows from (9), while identity (10) is just a a special case of Lemma 2.1 [1] applied for \(k=1\) and \(\delta_1=\Omega\).
As \(0\le A\le B\) for \(A,B\in \mathcal{C_{\infty}(H)}\) implies \( s_n^\theta(A)\le s_n^\theta(B)\) for all \(n\in \mathbb{N}\), as well as \(|\| A^\theta|\|\le |\| B^\theta|\|,\) then (12) follows.
Recall that, for a pair of random real variables \((Y,Z)\), its coefficient of correlation
\[\rho_{Y,Z}=\frac{| E(YZ)-E(Y)E(Z)|}{\sigma(Y)\sigma(Z)}= \frac{| E(YZ)-E(Y)E(Z)|}{ \sqrt{E(Y^2)-E^2(Y)} \sqrt{E(Z^2)-E^2(Z)}}\] always satisfies \(|\rho_{Y,Z}|\le 1.\) For square integrable functions \(f\) and \(g\) on \([0,1]\) and \(D(f,g)=\int_0^1f(t)g(t)\,d t- \int_0^1f(t)\,d t\int_0^1g(t)\,d t.\) Landau proved that \( | D(f,g)|\le \sqrt{D(f,f)D(g,g)}.\)The following result represents a generalization of Landau inequality in \(u.i.\) norm ideals [2] for Gel’fand integrals of \(o.v.\) functions with relative simplicity of its formulation.
Theorem 3. If \(\mu\) is a probability measure on \(\Omega\). Let both fields \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in\Omega}\) be in \(L^2(\Omega,\mu,\mathcal{B(H)})\) consisting of commuting normal operators and consider \[\sqrt{\,\int_\Omega|A_{t}|^2 -\left|\int_\Omega A_{t} d\mu(t)\right|^2}X \sqrt{\,\int_\Omega| B_{t}|^2 d\mu(t)-\left|\int_\Omega B_{t} d\mu(t)\right|^2},\] for some \(X\in B(H)\). Then \[\int_\Omega A_tX B_t d\mu(t)-\int_\Omega A_{t} dt X\!\!\int_\Omega B_{t} d\mu(t) \in C_{|\|.|\|}(H).\]
Proof. First, we have the following Korkine type identity for \(i.p.t.i\) transformers
Lemma 2. Let \(\mu\) (resp. \(\nu\)) be a probability measure on \(\Omega\) (resp. \(\mho\)). Further, let both families \(\{A_s,C_t\}_{(s,t)\in\Omega\times\mho}\) and \(\{B_s, D_t\}_{(s,t)\in\Omega\times\mho}\) consist of commuting normal operators and let \begin{equation}\sqrt{\,\int_\Omega|A_s|^2d\mu(s) \int_\mho|C_t|^2d\nu(t)-\left|\int_\Omega A_sd\mu(s)\int_\mho C_td\nu(t)\right|^2} X \sqrt{\,\int_\Omega| B_s|^2d\mu(s) \int_\mho|D_t|^2d\nu(t)-\left|\int_\Omega B_sd\mu(s)\int_\mho D_td\nu(t)\right|^2} \end{equation} be in \(\mathcal{C}_{|\|\cdot| |\|}(\mathcal{H})\) for some \(X\in \mathcal{B(H)}\). Then \begin{eqnarray*}\int_\Omega \int_\mho A_s C_tX B_s D_t\,d\mu(s)\,d\nu(t) -\int_\Omega A_s \,d\mu(s)\int_\mho C_t\,d\nu(t) X\!\!\int_\Omega B_s \,d\mu(s) \int_\mho D_t\,d\nu(t) \in\mathcal{C}_{|\|\cdot| |\|}(\mathcal{H}).\end{eqnarray*}
Proof. Apply Theorem 3 to the probability measure \(\mu\times\nu\) on \(\Omega\times\mho\) and families \((A_s C_t)_{(s,t)\in\Omega\times\mho}\) and \(( B_s D_t)_{(s,t)\in\Omega\times\mho}\) of normal commuting operators in \(L_G^2(\Omega\times\mho,d\mu\times\nu,\mathcal{B(H)}).\) The rest follows trivially.
Next, we consider Landau type inequality for \(i.p.t.i\) transformers in Schatten ideals for the Schatten \(p\)-norms.
Proposition 5. Let \(\mu\) be a probability measure on \(\Omega\) and \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in \Omega}\) be \(\mu\)-weak\({}^*\) measurable families of bounded Hilbert space operators such that \(\int_\Omega\left(\|A_t f\|^2+\|A_t^* f\|^2+\| B_t f\|^2+\|B_t^* f\|^2\right)d\mu(t)< \infty\) for all \(f\in \mathcal{H}\) and let \(p,q,r\ge1\) such that \(\dfrac1p=\dfrac1{2q}+\dfrac1{2r}\,\). Then for all \(X\in \mathcal{C}_p(\mathcal{H})\),
Proof. According to identity (14), applying Theorem 3.3 [1] to families \((\mathcal{A}_s-\mathcal{A}_t)_{(s,t)\in\Omega^2}\) and \((\mathcal{B}_s-\mathcal{B}_t)_{(s,t)\in\Omega^2}\) gives
The next result [1] is a special case of an abstract Hölder inequality presented in Theorem 3.1.(e) [1] for Cauchy-Schwarz inequality for \(o.v.\) functions in \(u.i.\) norm ideals.
Proposition 6. Let \(\mu\) be a measure on \(\Omega\). Further, let \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in \Omega}\) be \(\mu\)-weak\({}^*\) measurable in \(\mathcal{B(H)}\) such that \(|\int_\Omega|A_t|^2 d\mu(t)|^\theta\) and \(|\int_\Omega|B_t|^2 d\mu(t)|^\theta\) are in \(\mathcal{C}_{\||.|\|}\mathcal{H}\) for some \(\theta>0\) and for \(u.i.\) norm. Then we have \[ \|||\int_\Omega A_t^* B_t d\mu(t)\|||^\theta \|||\le \|||\int_\Omega A_t^* A_t d\mu(t)\|||^\theta \|||^\frac12 \|||\int_\Omega B_t^* B_t d\mu(t)\|||^\theta \|||^\frac12. \]
Proof. Take \(\Phi\) to be a \(s.g.\) function that generates \(u.i.\) norm \(\||\cdot\||\), \(\Phi_1=\Phi\), \(\Phi_2=\Phi_3=\Phi^{(2)}\) (2-reconvexization of \(\Phi\)), \(\alpha=2\theta\) and \(X=I\), and then apply Theorem 3.1 [1], we get our desired result.
Now, we give another generalization of Landau inequality for Gel’fand integrals of \(o.v.\) functions in \(u.i.\) norm ideals.
Theorem 4. If \(\mu\) is a probability measure on \(\Omega\), \(\theta>0\) and \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in \Omega}\) are as in Proposition 6, \(\mu\)-weak\({}^*\) measurable families of bounded Hilbert space operators such that \(\|||\int_\Omega|A_t|^2d\mu(t)\|||^\theta\) and \(\|||\int_\Omega|B_t|^2d\mu(t)\|||^\theta\) are in \(\mathcal{C}_{\||.|\|}\mathcal{H}\) for some \(\theta>0\) and for some \(u.i.\) norm \(\||\cdot\||\) we have \begin{eqnarray*} && \left\|\left|\int_\Omega A_t^* B_td\mu(t) -\int_\Omega A_t^*d\mu(t)\int_\Omega B_td\mu(t) \|||^\theta \right\|\right|^2\\ &&\le \||\int_\Omega \||| A_t \|||^2d\mu(t)- \|||\int_\Omega A_td\mu(t) \|||^2 \|||^\theta \||| \|||\int_\Omega \||| B_t \|||^2d\mu(t)- \|||\int_\Omega B_td\mu(t) \|||^2 \|||^\theta \||.\end{eqnarray*}
Proof. It suffices to invoke Proposition 6 to \(o.v.\) families \((A_s-A_t)_{(s,t)\in\Omega^2}\) and \((B_s-B_t)_{(s,t)\in\Omega^2}\) and use identity [7] to proof this result.
Now, we consider some interesting quantities that relate to norm inequalities. For bounded set of operators \(A=(\mathcal{A}_t)_{t\in\Omega}\), we see that the radius of the smallest disk that essentially contains its range is
\[r_\infty(A)=\inf_{A\in \mathcal{B(H)}}ess \sup_{t\in\Omega}\| A_t-A\|= \inf_{A\in \mathcal{B(H)}}\| A_t-A\|_\infty=\min_{A\in \mathcal{B(H)}}\| A_t-A\|_\infty.\] From the triangle inequality, we have \(\bigl|\|\mathcal{A}_t-A’\|-\|\mathcal{A}_t-A\|\bigr|\leq\|A’-A\|\), so the mapping \(A\to ess \sup_{t\in\Omega}\|A_t-A\|\) is nonnegative and continuous on \(\mathcal{B(H)}\). Since \((\mathcal{A}_t)_{t\in\Omega}\) is bounded field of operators, we also have \(\| A_{t}-A\|\to\infty\) when \(\|A\|\to\infty\), so this mapping attains minimum [5], and it actually attains at some \(A_0\in \mathcal{B(H)}\), which represents a center of the disk considered [6]. Any such field of operators is of finite diameter, therefore, we have that \(r_\infty(A)=ess \sup_{s,t\in\Omega}\| A_s-A_t\|,\) with the simple inequalities given as \(r_\infty(A)\le diam_\infty(A)\le 2r_\infty(A)\) relating those quantities. For such fields of operators we can now state the following stronger version of Grüss inequality [2].Lemma 3. Let \(\mu\) be a \(\sigma\)-finite measure on \(\Omega\) and let \(A=(\mathcal{A}_t)_{t\in\Omega}\) and \(B=(\mathcal{B}_t)_{t\in\Omega}\) be \([\mu]\) a.e. bounded fields of operators. Then, for all \(X\in \mathcal{C_{|\|.|\|}(H)}\), \( \sup_{\mu(\delta)>0}\||\frac1{\mu(\delta)}\int_\delta\mathcal{A}_tX\mathcal{B}_t d\mu(t) – \frac1{\mu(\delta)}\int_\delta\mathcal{A}_t d\mu(t) \,X \frac1{\mu(\delta)}\int_\delta\mathcal{B}_t d\mu(t) |\|\le \min_{i} \mathcal{P}_{i}\cdot\|| X\||.\) (Here \(\sup\) is taken over all measurable sets \(\delta\subseteq\Omega\) such that \(0< \mu(\delta)< \infty\)).
Lemma 3 has the following immediate implication when \((\mathcal{A}_t)_{t\in\Omega}\) and \((\mathcal{B}_t)_{t\in\Omega}\) are bounded fields of self-adjoint operators.
Theorem 5. If \(\mu\) is a probability measure on \(\Omega\) and \(C,D,E,F\) be bounded self-adjoint operators. Also, let \((\mathcal{A}_t)_{t\in\Omega}\) and \((\mathcal{B}_t)_{t\in\Omega}\) be bounded self-adjoint fields satisfying \(C\le\mathcal{A}_t\le D\) and \(E\le\mathcal{B}_t\le F\) for all \(t\in\Omega\). Then for all \(X\in \mathcal{C_{|\|.|\|}(H)}\), we have
Proof. As \(\frac{C-D}2\le\mathcal{A}_t-\frac{C+D}2\le\frac{D-C}2\) for every \(t\in\Omega\), then \begin{eqnarray*} ess \sup_{t\in\Omega}\| \mathcal{A}_t-\frac{C+D}2\|= ess \sup_{t\in\Omega}\sup_{\| f\|=1}\||\langle\mathcal{A}_t-\frac{C+D}2 \| f,f\rangle|\le \sup_{\| f\|=1}|\langle\frac{D-C}2 f,f\rangle|= \frac{\| D-C\|}2, \end{eqnarray*} which implies \(r_\infty(A)\le \frac{\| D-C\|}2,\) and similarly \(r_\infty(B)\le\frac{\| F-E\|}2.\) Thus (18) follows directly.
In case of \(\mathcal{H}=\mathbb{C}\) and \(\mu\) being the normalized Lebesgue measure on \([a,b]\) (i.e. \(d\,\mu(t)=\frac{dt}{b-a}\)), then (1) follows from Theorem 5. This special case also confirms the sharpness of the constant \(\frac14\) in the inequality (18).
Lastly, we consider, the Grüss type inequality for elementary operators in the example below.
Example 1. Let \(A_1, \ldots, A_n\), \(B_1, \ldots, B_n\), \(C, D, E\) and \(F\) be bounded linear self-adjoint operators acting on a Hilbert space \(\mathcal{H}\) such that \(C\le A_i\le D\) and \(E\le B_i\le F\) for all \(i=1,2,\cdots,n\), then for arbitrary \(X\in\mathcal{C}_{\||.|\|}\mathcal{H}\), we have \begin{eqnarray} \nonumber%\label{mainc} \|| \frac1n\sum_{i=1}^n A_i XB_i-\frac1{n^2}\sum_{i=1}^nA_i\, X\sum_{i=1}^nB_i\|| \leq \frac{\|D-C\|\|F-E\|}4 \|| X\||. \end{eqnarray} Indeed, it is sufficient to prove that the elementary operator is normally represented and that Grüss type inequality holds for it [3].
In the next section, we dedicate our effort to the applications of this study in other fields. We consider quantum theory in particular, whereby, we describe the application in quantum chemistry and quantum mechanics.
The quantum mechanics deals with commutator approximation. The discussions of approximation by commutators \(AX-XA\) or by generalized commutator \(AX-XB\) originates from quantum theory. For instance, the Heisenberg uncertainly principle may be mathematically deduced as saying that there exists a pair \(A,X\) of linear operators and a non-zero scalar \(\alpha\) for which \(AX – XA = \alpha I\). A natural question immediately arises: How close can \(AX – XA\) be to the identity? In [3], it is discussed that if \(A\) is normal, then, for all \(X \in B(H)\), \(||I – (AX – XA)|| \geq ||I||.\) In the inequality here, the zero commutator is a commutator approximate in \(B(H)\).