Positive Invertible Operators Research Articles

Sphere bundle over the set of inner products in a Hilbert space

Let (H,〈,〉) be a complex Hilbert space and B(H) the space of bounded linear operators in H. Any other equivalent inner product in H is of the form 〈f,g〉A=〈Af,g〉 (f,g∈H) for some positive invertible operator A∈B(H). In this paper we study the bundle M which consist of the unit sphere {f∈H:〈f,f〉A=1} over each (equivalent) inner product 〈,〉A, which due to the observation above can be definedM={(A,f)∈B(H)×H:A is positive and invertible and 〈Af,f〉=1}. We prove that M is a complemented submanifold of the Banach space B(H)×H and a homogeneous space of the Banach-Lie group G(H)⊂B(H) of invertible operators. We introduce a reductive structure in M, and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of M, for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C⁎-algebra A⊂B(H).

Maximal ergodic inequalities for some positive operators on noncommutative Lp$L_p$‐spaces

AbstractIn this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative ‐spaces for a fixed , which particularly applies to positive isometries, the convex hull of positive Lamperti contractions, and also the power‐bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces . As an unexpected consequence, we deduce a completely bounded version of Ackoglu's theorem by making use of Ackoglu's original dilation. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge–Le Merdy [J. Funct. Anal. 249 (2007), no. 1, 220–252.], still satisfy the maximal ergodic inequalities. Together with the class of power‐bounded positive invertible operators considered by Hong–Liao–Wang [Duke Math. J. (2020). In press], we thereby verify the noncommutative Ackoglu ergodic theorem for all the positive operators that have naturally appeared in the literature up to the moment of writing. Based on the noncommutative Calderón transference principle established in [Duke Math. J. (2020). In press], the general pattern of the demonstration follows from the classical one due to Kan, but several new ideas are absolutely necessary in the noncommutative setting. Let us just mention three of them: the argument for the maximal ergodic theorem for positive isometries is highly nontrivial compared to the classical case; the novel simultaneous dilation property is indispensable to get the ergodic theorem for the convex hull of positive Lamperti contractions; numerous adjustments are truly needed in this new setting to conclude a desired structural theorem for positive doubly Lamperti operators since the orthogonal relations of operator algebras are completely different from those in classical measure theory.

Logarithmic mean of positive invertible operators

Logarithmic mean of positive invertible operators

Mixture and interpolation of the parameterized ordered means

Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means.

Ergodic theorems for Cesàro bounded operators in L1

We consider a positive invertible Lamperti operator T with positive inverse. Our result concerns the averages Ak,n, 0≤k,n∈Z, and the ergodic maximal operatorMTf=supk,n≥0⁡|Ak,nf|=supk,n≥0⁡|1k+n+1∑j=−knTjf|, the ergodic Hilbert transform defined by HTf(x)=limn→∞⁡∑j=1n1j(Tjf(x)−T−jf(x)) and the ergodic power functions defined by Pr,Tf(x)=(∑n=0∞|An+1,0f(x)−An,0f(x)|r+|A0,n+1f(x)−A0,nf(x)|r)1/r, for 1<r<+∞. Several authors proved that if the averages are uniformly bounded in Lp, 1<p<∞, then these operators are bounded in Lp[23,24,27,28,21]. Regarding the case p=1, Gillespie and Torrea [12] showed that this condition is not sufficient to assure MT is of weak type (1,1). We provide two sufficient conditions that recall the assumptions in the Dunford-Schwartz theorem: if the averages are uniformly bounded in L1 and in L∞ then MT, HT and Pr,T apply L1 into weak-L1 and the corresponding sequences of functions in L1 converge a.e. and in measure. Furthermore, we reach the same conclusions by assuming a weaker condition: we replace the uniform boundedness of the averages An,n in L∞ by the assumption that, for a fixed p∈(1,∞), the averages associated with a modified operator Tp, related with T, are uniformly bounded in Lp. We end the paper showing examples of nontrivial operators satisfying the assumptions of the two main results.

Expansive operators which are power bounded or algebraic

Given Hilbert space operators $P,T\in B(\H), P\geq 0$ invertible, $T$ is $(m,P)-$ expansive (resp., $(m,P)-$ isometric) for some positive integer $m$ if $\triangle_{T^*,T}^m(P)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right){T^*}^jPT^j\leq 0$ (resp., $\triangle_{T^*,T}^m(P)=0$). An $(m,P)-$ expansive operator $T$ is power bounded if and only if it is a $C_{1\cdot}-$ operator which is similar to an isometry and satisfies $\triangle_{T^*,T}^n(Q)=0$ for some positive invertible operator $Q\in B(\H)$ and all integers $n\geq 1$. If, instead, $T$ is an algebraic $(m,I)-$ expansive operator, then either the spectral radius $r(T)$ of $T$ is greater than one or $T$ is the perturbation of a unitary by a nilpotent such that $T$ is $(2n-1, I)-$ isometric for some positive integers $m_0 \leq m$, $m_0$ odd, and $n \geq \frac{m_0 +1}{2}$.

Some Multiplicative Inequalities for Heinz Operator Mean

In this paper we obtain some new multiplicative inequalities for Heinz operator mean. 1. Introduction Throughout this paper A; B are positive invertible operators on a complex Hilbert space (H; h ; i) : We use the following notations for operators and 2 [0; 1] Ar B := (1 )A+ B; the weighted operator arithmetic mean, and A] B := A 1=2 A BA 1=2 A; the weighted operator geometric mean [14]. When = 1 2 we write ArB and A]B for brevity, respectively. De ne the Heinz operator mean by H (A;B) := 1 2 (A] B +A]1 B) : The following interpolatory inequality is obvious (1.1) A]B H (A;B) ArB for any 2 [0; 1]: We recall that Specht’s ratio is de ned by [16] (1.2) S (h) := 8> >: h 1 h 1 e ln h 1 h 1 if h 2 (0; 1) [ (1;1) ; 1 if h = 1: It is well known that limh!1 S (h) = 1; S (h) = S 1 h > 1 for h > 0; h 6= 1. The function is decreasing on (0; 1) and increasing on (1;1) : The following result provides an upper and lower bound for the Heinz mean in terms of the operator geometric mean A]B : Theorem 1 (Dragomir, 2015 [6]). Assume that A; B are positive invertible operators and the constants M > m > 0 are such that (1.3) mA B MA: 1991 Mathematics Subject Classi cation. 47A63, 47A30, 15A60, 26D15, 26D10.

Further inequalities involving the weighted geometric operator mean and the Heinz operator mean

ABSTRACT In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean where is a real number and A, B are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to with a high rate of convergence. Some relationships involving are deduced. Numerical examples illustrating the theoretical results are also discussed.

On Functional Representations of Positive Hilbert Space Operators

In this paper we consider some faithful representations of positive Hilbert space operators on structures of nonnegative real functions defined on the unit sphere of the Hilbert space in question. Those representations turn order relations between positive operators to order relations between real functions. Two of them turn the usual Lowner order between operators to the pointwise order between functions, another two turn the spectral order between operators to the same, pointwise order between functions. We investigate which algebraic operations those representations preserve, hence which kind of algebraic structure the representing functions have. We study the differences among the different representing functions of the same positive operator. Finally, we introduce a new complete metric (which corresponds naturally to two of those representations) on the set of all invertible positive operators and formulate a conjecture concerning the corresponding isometry group.

Asymmetric Choi–Davis inequalities

ABSTRACT Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such that and hold under some mild operator convex conditions and some positive numbers r. Further, we show that if is operator concave, then In addition, we give some counterparts to the asymmetric Choi–Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin–Ricard and Furuta.

Estimates of Ando–Hiai-type inequalities on operator power means

ABSTRACT We aim to give Ando–Hiai-type ratio inequalities for an operator power mean and its reverse ones which represent a relation between and for r>0 under k-tuple of positive invertible operators and . As applications, we consider complementary inequalities related to the information monotonicity property via any unital positive linear map. In particular, we state some results for the Karcher means by .

Noncommutative maximal ergodic inequalities associated with doubling conditions

In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong-Liao-Wang. Our strategy essentially relies on various structural characterizations and dilation properties associated with Lamperti operators, which are of independent interest. More precisely, we give a structural description of Lamperti operators in the noncommutative setting, and obtain a simultaneous dilation theorem for Lamperti contractions. As a consequence we establish the maximal ergodic theorem for the strong closure of the convex hull of corresponding family of positive contractions. Moreover, in conjunction with a newly-built structural theorem, we also obtain the maximal ergodic inequalities for positive power bounded doubly Lamperti operators. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge-Le Merdy, still satisfy the maximal ergodic inequalities. We also discuss some other examples, showing sharp contrast to the classical situation.

On Schatten restricted norms

We consider norms on a complex separable Hilbert space such that $\langle a\xi,\xi\rangle\leq\|\xi\|^2\leq\langle b\xi,\xi\rangle$ for positive invertible operators $a$ and $b$ that differ by an operator in the Schatten class. We prove that these norms have unitarizable isometry groups, our proof uses a generalization of a fixed point theorem for isometric actions on positive invertible operators. As a result, if the isometry group does not leave any finite dimensional subspace invariant, then the norm must be Hilbertian. That is, if a Hilbertian norm is changed to a close non-Hilbertian norm, then the isometry group does leave a finite dimensional subspace invariant. The approach involves metric geometric arguments related to the canonical action of the group on the non-positively curved space of positive invertible Schatten perturbations of the identity .

VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

AbstractFor an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Functions preserving operator means

Let \(\sigma\) be a non-trivial operator mean in the sense of Kubo and Ando, and let \(OM_+^1\) be the set of normalized positive operator monotone functions on \((0, \infty )\). In this paper, we study the class of \(\sigma\)-subpreserving functions \(f\in OM_+^1\) satisfying $$\begin{aligned} f(A\sigma B) \le f(A)\sigma f(B) \end{aligned}$$for all invertible positive operators A and B. We provide some criteria for f to be trivial, i.e., \(f(t)=1\) or \(f(t)=t\). We also establish characterizations of \(\sigma\)-preserving functions \(f\in OM_+^1\) satisfying $$\begin{aligned} f(A\sigma B) = f(A)\sigma f(B) \end{aligned}$$for all invertible positive operators A and B. In particular, when \(\lim _{t\rightarrow 0} (1\sigma t) =0\), the function \(f\in OM_+^1\backslash \{1,t\}\) preserves \(\sigma\) if and only if f and \(1\sigma t\) are representing functions for a weighted harmonic mean.

SPECTRAL ORDER OF OPERATORS AND FURUTA INEQUALITY

By virtue of Furuta inequality, we show some characterizations of the spectral order of positive operators on a Hilbert space from the viewpoint of the Kan- torovich type inequality and the Riccati equation. Let A and B be positive operators on a Hilbert space. Then the spectral order AB holds if and only if there exist a unique positive contraction Tp,u such that Tp,uA p+u 2 Tp,u = A up 4 B p A up 4 for all p ≥ 0 and u ≥ 0. This form interpolates the chaotic order, δ-order and the usual order continuously. 1 Introduction. An operator means a bounded linear operator on a Hilbert space H. An operator A is said to be positive ( denoted by A ≥ 0) if (Ax, x) ≥ 0 for all x ∈ H.I n particular, we denote by A> 0i fA is positive and invertible. The usual order A ≥ B for selfadjoint operators A and B is defined by A − B ≥ 0. Let A and B be positive invertible operators. We introduce a function order A ≥f B for a real valued continuous function f on (0, ∞) defined by A ≥f B ⇐⇒ f(A) ≥ f(B).

GEOMETRIC MEANS OF POSITIVE OPERATORS II

Based on Ricatti equation XA 1 X = B for two (positive invertible) operators A and B which has the geometric mean A)B as its solution, we consider a cubic equation X(A)B) 1 X(A)B) 1 X = C for A;B and C: The solution X = (A)B)) 1 C is a candidate of the geometric mean of the three operators. However, this solution is not invariant under permutation unlike the geometric mean of two operators. To supply the lack of the property, we adopt a limiting process due to Ando-Li-Mathias. We dene reasonable geometric means of k operators for all integers k 2 by induction. For three positive operators, in particular, we dene the weighted geometric mean as an extension of that of two operators.

Operator means of probability measures

Let P be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion of operator means of probability measures on P, in parallel with Kubo and Ando's definition of two-variable operator means, and show that every operator mean is contractive for the ∞-Wasserstein distance. By means of a fixed point method we consider deformation of such operator means, and show that the deformation of any operator mean becomes again an operator mean in our sense. Based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures.

Some remarks on Tsallis relative operator entropy

This paper intends to give some new estimates for Tsallis relative operator entropy ${{T}_{v}}\left( A|B \right)=\frac{A{{\natural}_{v}}B-A}{v}$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $J \subset (0,\infty)$. We prove for any $v\in \left[ -1,0 \right)\cup \left( 0,1 \right]$, $$ (\ln_v t)A+\left( A{{\natural}_{v}}B+tA{{\natural}_{v-1}}B \right)\le {{T}_{v}}\left( A|B \right) \le (\ln_v s)A+{{s}^{v-1}}\left( B-sA \right) $$ where $s,t\in J$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Meanwhile, some related and new results are also established. In particular, the monotonicity for Tsallis relative operator entropy is improved. Furthermore, we introduce the exponential type relative operator entropies which are special cases of the perspective and we give inequalities among them and usual relative operator entropies.

Concavity of some operator functions and Fréchet differential mappings

By using operator perspectives of regular operator mappings we obtain some operator concave (convex) functions and derive the concavity (convexity) of some trace functions associated with the Fréchet differential mapping of certain power functions. Especially, we explore the concavity of the Fréchet differential mapping x→dg(x)⁎df(x)−1 in positive definite invertible operators with f(t)=tp for 0<p≤1 and g(t)=tq for p≤q≤p+1, and also with f(t)=log⁡t and g(t)=tq for 0<q≤1.