- Research
- Open Access
Functional version for Furuta parametric relative operator entropy
- Mustapha Raïssouli^{1, 2} and
- Shigeru Furuichi^{3}Email author
https://doi.org/10.1186/s13660-018-1804-x
© The Author(s) 2018
- Received: 3 April 2018
- Accepted: 6 August 2018
- Published: 15 August 2018
Abstract
Functional version for the so-called Furuta parametric relative operator entropy is here investigated. Some related functional inequalities are also discussed. The theoretical results obtained by our functional approach immediately imply those of operator versions in a simple, fast, and nice way.
Keywords
- Operator inequalities
- Functional inequalities
- Operator entropies
- Convex analysis
MSC
- 46N10
- 46A20
- 47A63
- 47N10
- 39B62
- 52A41
1 Introduction
The fundamental goal of this paper is to give an extension of \(S_{p}(A|B)\) when the operator variables A and B are (convex) functionals. Some functional relationships and inequalities are provided as well. The related operator versions are deduced in a fast and nice way.
2 Functional extensions
The previous operator concepts have been extended from the case that the variables are positive operators to the case that the variables are convex functionals, see [9].
3 Needed tools
As usual we denote by \(\Gamma_{0}(H)\) the cone of all functionals \(f\in \tilde{\mathbb{R}}^{H}\) that are convex, lower semi-continuous, and proper (i.e., not identically equal to +∞). It is well known that \(f^{**}:=(f^{*})^{*}\leq f\) for any \(f\in \tilde{\mathbb{R}}^{H}\) and \(f\in \Gamma_{0}(H)\) if and only if \(f=f^{**}:=(f^{*})^{*}\). Moreover, \(x^{*}\in \partial f(x)\) always implies \(x\in \partial f ^{*}(x^{*})\), with reversed implication provided that \(f\in \Gamma _{0}(H)\).
For the sake of clearness and simplicity for the reader, we state the following example illustrating the previous concepts.
Example 3.1
- (i)Assume that \(A\in {\mathcal{B}}^{+*}(H)\). Then \(f_{A}\) is convex and G-differentiable on H, and soThe coefficient \(1/2\) appearing in \(f_{A}\) enjoys a symmetry role in the aim to have$$ \forall x\in H\quad \partial f_{A}(x)=\bigl\{ \nabla f_{A}(x)\bigr\} =\{Ax \}. $$$$ (f_{A})^{*}\bigl(x^{*}\bigr)=(1/2)\bigl\langle A^{-1}x^{*},x^{*}\bigr\rangle \quad \mbox{for all } x^{*}\in H, \mbox{ or in short } (f_{A} )^{*}=f _{A^{-1}}. $$
- (ii)
For any \(A,B\in {\mathcal{B}}(H)\), it is easy to check that \(f_{A}\pm f_{B}=f_{A\pm B}\) and \(f_{A}(Bx)=f_{BAB}(x)\) for any \(x\in H\).
The following result, which will be needed later, has been proved in [11].
Theorem 3.2
- (i)The inequalityholds true for all \(x\in \operatorname{int}(\operatorname{dom}f)\).$$ \sup_{x^{*}\in \partial f(x)} \bigl(f^{*}-g^{*} \bigr) \bigl(x^{*}\bigr)\leq {\mathcal{S}}(f/g) (x) \leq (g-f) (x) $$(3.1)
- (ii)If f is moreover G-differentiable at x, then we have$$ f^{*} \bigl(\nabla f(x) \bigr)-g^{*} \bigl( \nabla f(x) \bigr)\leq {\mathcal{S}}(f/g) (x) \leq (g-f) (x). $$(3.2)
As explained in [11], (3.1), as well as (3.2), is a functional extension of (1.3) from positive operators to convex functionals.
We have the following result summarizing the elementary properties of \({\mathcal{T}}_{p}^{*}(f|g)\).
Proposition 3.3
- (i)For any \(p\in [0,1)\), one has$$ {\mathcal{T}}_{1-p}^{*}(g|f)=\frac{ ({\mathcal{G}}_{p}(f,g) )^{*}-g ^{*}}{1-p}. $$
- (ii)For all \(p\in (0,1]\), the left-hand side of the inequalityholds for any \(x^{*}\in H\), while the right-hand side holds for \(x^{*}\) such that \(g^{*}(x^{*})=+\infty \) or \(x^{*}\in \operatorname{dom}f^{*}\).$$ \frac{ ({\mathcal{A}}_{p}(f,g) )^{*}(x^{*})-f^{*}(x^{*})}{p} \leq {\mathcal{T}}_{p}^{*}(f|g) \bigl(x^{*}\bigr)\leq g^{*}\bigl(x^{*} \bigr)-f^{*}\bigl(x^{*}\bigr) $$
Proof
(i) Follows from (3.3) with the relation \({\mathcal{G}}_{p}(f,g)= {\mathcal{G}}_{1-p}(g,f)\).
Proposition 3.4
Proof
4 Functional version of \(S_{p}(A|B)\)
As already pointed out before, our aim here is to give an analog of \(S_{p}(A|B)\) when the operator arguments A and B are (convex) functionals f and g, respectively. Such an analog seems to be hard to define from (1.4) since (1.4) involves the product of operators whose analogs for functionals are not known yet. For this, we need to state the following result.
Theorem 4.1
Proof
The second equality can be proved in a similar manner. □
Now, to give a functional version of \(S_{p}(A|B)\), we use (4.1) which is more appropriate for our aim since (4.1) involves only operator concepts (relative operator entropy and operator geometric mean) whose functional extensions are already done. Taking into account a symmetric character between p and \(1-p\) in our desired definition, we then put the following.
Definition 4.2
As a first result we state the following.
Proposition 4.3
Proof
Equality (4.3) is immediate from (4.2). However, we mention that (4.4) is not immediate from (4.2) since our involved functionals could take the value +∞. Indeed, we pay attention to the fact that, if \(\phi ,\psi \in \tilde{\mathbb{R}}^{H}\), the equality \(\phi -\psi =-(\psi -\phi )\) is not always true unless \(\operatorname{dom}\phi \cup \operatorname{dom}\psi =H\). For this reason, we have assumed in our statement that \(\operatorname{dom}f=\operatorname{dom}g=H\) in the aim to guarantee that \({\mathcal{H}}_{t} ({\mathcal{G}}_{p}(f,g),g )\) or \({\mathcal{H}}_{t} ({\mathcal{G}}_{p}(f,g),f )\) is with finite values. With this, (4.4) can be deduced from (4.2) when we refer to the relationship \({\mathcal{G}}_{p}(\phi ,\psi )= {\mathcal{G}}_{1-p}(\psi ,\phi )\) valid for any \(\phi ,\psi \in \tilde{\mathbb{R}}^{H}\) and \(p\in [0,1]\). □
A connection between the functional parametric entropy \({\mathcal{S}} _{p}(f|g)\) and the operator parametric entropy \(S_{p}(A|B)\) is expressed by the following result.
Proposition 4.4
Proof
Relationship (4.5) justifies that \({\mathcal{S}}_{p}(f|g)\) is a reasonable extension of \(S_{p}(A|B)\), from operators to functionals, in the sense of (2.5).
We now are in a position to state the following main result.
Theorem 4.5
Proof
Since \(\mathrm{int} (\operatorname{dom}{\mathcal{G}}_{p}(f,g) )\neq \emptyset \), then \(\partial {\mathcal{G}}_{p}(x)\neq \emptyset \) for any \(x\in \operatorname{int} (\operatorname{dom}{\mathcal{G}}_{p}(f,g) )\).
Remark 4.6
It is worth mentioning that the condition \(\mathrm{int} (\operatorname{dom}{\mathcal{G}}_{p}(f,g) )\neq \emptyset \) is satisfied if \(\mathrm{int} (\operatorname{dom}f\cap \operatorname{dom}g )\neq \emptyset \) since \(\operatorname{dom}f\cap \operatorname{dom}g\subset \operatorname{dom}{\mathcal{G}}_{p}(f,g)\).
Corollary 4.7
Proof
Since \({\mathcal{G}}_{p}(f,g)\) is G-differentiable at x, then \(\partial {\mathcal{G}}_{p}(f,g)(x)=\{\nabla {\mathcal{G}}_{p}(f,g)(x) \}\). Substituting this in (4.6) and using the definition of the point-wise order, we immediately obtain the desired inequalities. □
The operator version of the above theorem (and corollary) reads as follows.
Corollary 4.8
Proof
Combining Corollary 4.7, Proposition 3.4, and Example 3.1,(ii), we obtain the desired operator inequalities after simple manipulations. The details are simple and therefore omitted. □
Corollary 4.8 gives the relation between Furuta parametric relative operator entropy and Tsallis relative operator entropy in a more general setting than the result in [4, Theorem 2.3].
Declarations
Acknowledgements
The authors thank anonymous referees for giving valuable comments and suggestions to improve our manuscript.
Funding
The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 16K05257.
Authors’ contributions
The work presented here was carried out in collaboration between all authors. All authors have contributed to, checked, read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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