Functional version for Furuta parametric relative operator entropy

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.

Otherwise, the relative operator entropy S(A|B) and the Tsallis relative operator entropy T p (A|B) are, respectively, defined by (see [2,3,6]) The following double inequality is known in the literature: In [5], Furuta introduced a parametric extension of S(A|B) as follows: In fact, S p (A|B) was introduced in [5] for any real number p, but here we restrict ourselves to the case p ∈ [0, 1].
As pointed out in [5], it is not hard to see that S 0 (A|B) = S(A|B), S 1 (A|B) = -S(B|A) and S p (A|B) = -S 1-p (B|A).
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.

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].
LetR H be the extended space of all functionals defined from H into R ∪ {+∞}. Let f , g ∈R H be two given functionals (convex or not) and p ∈ (0, 1). The expressions are called, by analogy, the weighted functional arithmetic mean, the weighted harmonic mean, and the weighted geometric mean of f and g, respectively. Here, the notation f * refers to the Fenchel conjugate of f defined by (2.2) For p = 1/2, we will denote the previous functional means by A(f , g), H(f , g) and G(f , g), respectively. We extend these means on the whole interval [0, 1] by setting: We mention that here we adopt the conventions 0.(+∞) = +∞ and (+∞) -(+∞) = +∞, as usual in convex analysis [1,8]. With this, relations (2.3) are not immediate from their related functional means (2.1) since the involved functionals f and/or g can take the value +∞.
For the same reason, analogous relationships of (1.1) for the previous functional means are also valid, i.e., In fact, the first two relations are immediate from their definitions, and for the third one, there is a detailed proof in [12]. Also, the analog of (1.2), i.e., with the convention +∞ -(+∞) = +∞ as already pointed before. The double inequality (2.4) implies that the three involved functional means are with finite values whenever f and g are so.
In the earlier papers [9] and [10] we extended S(A|B) and T p (A|B) from operators to (convex) functionals, respectively, as follows: The previous functional concepts were constructed as extensions of their related operator versions in the following sense: if O(A, B) is one of the previous operator concepts, its functional extension F(f , g) is such that where the notation f T , for any T ∈ B(H), refers to the quadratic function generated by the operator T, i.e., f T ( 3 Needed tools for any x * ∈ H. As supremum of a family of affine (so convex) functions, f * is always convex even if f is not. The conjugate map f − → f * is point-wise decreasing and convex. That is, As it is well known, ∂f (x) is a (possibly empty) convex and closed set.
In the case where ∂f (x) = ∅, we have the equivalence: As usual we denote by 0 (H) the cone of all functionals f ∈R H that are convex, lower semi-continuous, and proper (i.e., not identically equal to +∞). It is well known that f * * : For the sake of clearness and simplicity for the reader, we state the following example illustrating the previous concepts. (i) Assume that A ∈ B + * (H). Then f A is convex and G-differentiable on H, and so The coefficient 1/2 appearing in f A enjoys a symmetry role in the aim to have The following result, which will be needed later, has been proved in [11].

Theorem 3.2 Let f ∈ • (H) be such that int(dom f ) is nonempty. Then
As explained in [11], (3.1), as well as (3.2), is a functional extension of (1.3) from positive operators to convex functionals.
For the sake of simplicity for the reader, we need to introduce an auxiliary notation. For f , g ∈R H and p ∈ [0, 1], we set We have the following result summarizing the elementary properties of T * p (f |g).

Proposition 3.3 The following assertions hold:
(i) For any p ∈ [0, 1), one has (ii) For all p ∈ (0, 1], the left-hand side of the inequality holds for any x * ∈ H, while the right-hand side holds for x * such that g * (x * ) = +∞ or x * ∈ dom f * .

.4) we obtain by taking the conjugate side by side
Remarking that we then deduce the desired result.
This, with the fact that αf T = f αT and f Tf S = f T-S for any α ∈ R and T, S ∈ B(H), immediately yields the desired result.

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.
Proof Indeed, we have the property for any A, B ∈ B * (H) and any invertible operator T ∈ B(H) by using Kubo-Ando theory [7] and the integral form We thus have the first equality as since S(A|I) = -A log A for any A ∈ B + * (H).
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 1p in our desired definition, we then put the following. Definition 4.2 Let f , g ∈R H and p ∈ [0, 1]. We set with S 0 (f |g) = S(f |g) and S 1 (f |g) = -S(g|f ).
As a first result we state the following.

Proposition 4.3 Let f , g ∈R H . Then we have
Further, if dom f = dom g = H, then the equality holds for any p ∈ (0, 1). A connection between the functional parametric entropy S p (f |g) and the operator parametric entropy S p (A|B) is expressed by the following result. (4.5) Proof By (4.2), with (2.5) and (4.1), we have 2p .
This, with similar arguments as in the proof of Proposition 3.4, implies the desired result.
Relationship (4.5) justifies that S p (f |g) is a reasonable extension of S p (A|B), from operators to functionals, in the sense of (2.5).
Proof Since G p (f , g) is G-differentiable at x, then ∂G p (f , g)(x) = {∇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. 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].