Some Inequalities for Norms of Commutators

A short proof is given based on C*-algebra theory for the well-known theorem that if S and Tare bounded selfadjoint operators on a Hubert space such that 0^5^T then Sai|7a for each0^a 0; and S+e/and T+el are both invertible. Since (S+eI)x converges to Sx in norm when e-0 for each a>0, and since the positive operators in B(%>) form a norm closed set, it suffices to prove the theorem assuming that S and T are invertible. (The case a=0 can be verified directly, since S° is the range projection of 5.) Let E denote the set of exponents a in [0, 1] for which the function t—t is operator monotone. Trivially 0e£ and 1 eE. Since the function a-S is continuous from [0, 1] to F(§) in the norm topology we see that Fis a closed set. The proof will be complete when we show that Fis convex. Take a and s in E. Then S°^TX; hence r^^F72^/. It follows that \\S/2T-'*\\£l. Similarly \\Ss'2T-s/2\\^\. With P(A) the spectral radius of an operator A we have p(AB) = p(BA). Therefore /j*-U+s)/lMa+s)/2j--<+s1li) _ D('r'.a-s)li'j—iix+s'iHc{a+s)lij-l.!i+s)H-Y-t.!ii-s)li _ „(T—f/!c(»+j!)/2T-a/2 <; II j—0/2c(«t+/»/2<T--«/2t| <; \\T-pl2Ssl2 ||S°,/2T_,I/21| 1. Received by the editors April 27, 1972. AMS 1970 subject classifications. Primary 47B15; Secondary 46L05.

A capital letter means a bounded liner operator on a complex Hilbert space H , and B(H) denotes the algebra of all bounded linear operators on H equipped with operator norm ‖ · ‖ . An operator T is said to be positive (denoted by T 0) if (Tx,x) 0 for all x∈H and an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. Chaotic order is defined by logA logB for strictly positive operators A and B . A continuous real-valued function f defined on an interval J is called operator monotone if A B implies that f (A) f (B) for all self-adjoint operators A , B with spectra in J . The well known celebrated Lower-Heinz inequality asserts that if A B 0, then Aα Bα for any α ∈ [0,1] , which means that t → tα is operator monotone. Another well known example of operator monotone function is t → log t on (0,∞) . Recently, Furuta [5] showed that if A > B , then f (Aα ) > f (Bα) for all α ∈ (0,1] and if logA > logB , then there exists β ∈ (0,1] such that f (Aα) > f (Bα) for all α ∈ (0,β ] , where f (t) is a non-constant operator monotone function on [0,∞) . Comprehensive survey on related order preserving operator inequalities were given in [2,4]. In this paper, we will obtain an improvement which leads to the result of Moslehian and Najafi [6].

The main aim of this survey article is to present recent developments of matrix versions of the arithmetic–geometric mean inequality. Among others, we show improvements and generalizations of the arithmetic–geometric mean inequality for unitarily invariant norms via the Hadamard product, and for singular values via the operator monotone functions.

Loewner's theorem in several variables

An operator monotone function [formula omitted] and strictly chaotic order

We try to find a continuous functionu defined on a real right half-line with the range (0, ∞) such thatu −1 is operator monotone. We then look for another functionv such thatv(u −1) is operator monotone, namely,u(A)≦u(B) impliesv(A)≦v(B) for self-adjoint operatorsA andB.

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

Here we consider and study very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions. These are acting on the space of real fractionally differentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related fractional stochastic Shisha-Mond type inequalities of \(L^{q}\)-type \(1\le q\,<\infty \) and corresponding fractional stochastic Korovkin type theorems. These are regarding the stochastic q-mean fractional convergence of a sequence of stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are produced with rates and are given via the fractional stochastic inequalities involving the stochastic modulus of continuity of the \(\alpha \)th fractional derivatives of the engaged stochastic process, \(\alpha >0\), \(\alpha \notin \mathbb {N}\). The impressive fact is that the basic real Korovkin test functions assumptions are enough for the conclusions of our fractional stochastic Korovkin theory. We give applications to stochastic Bernstein operators. See also [10].

The subject is the applications of the use of quasi-entropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the f -divergence, monotone metrics, covariance and the \chi^2 -divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

The geometric separability probability of composite quantum systems has been extensively studied in the recent decades. One of the simplest but strikingly difficult problem is to compute the separability probability of qubit–qubit and rebit–rebit quantum states with respect to the Hilbert–Schmidt measure. A lot of numerical simulations confirm the and conjectured probabilities. We provide a rigorous proof for the separability probability in the real case and we give explicit integral formulas for the complex and quaternionic case. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit–qubit (and qubit-qutrit) states of the form of depends on (on single qubit subsystems), moreover it depends only on the Bloch radii (r) of D and it is constant in r. Using the Peres–Horodecki criterion for separability we give a mathematical proof for the probability and we present an integral formula for the complex case which hopefully will help to prove the probability, too. We prove Milz and Strunz’s conjecture for rebit–rebit and qubit–qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function is also studied in detail. We show that even in this setting Milz and Strunz’s conjecture holds true and we give an integral formula for separability probability according to this measure.

Let $\sigma$ be an operator mean and $f$ a non-constant operator monotone function on $(0,\infty)$ associated with $\sigma$. If operators $A, B$ satisfy $0\le A \le B$, then it holds that $Y \sigma (tA+X) \le Y \sigma (tB+X)$ for any non-negative real number $t$ and any positive, invertible operators $X,Y$. We show that the condition $ Y \sigma (tA+X) \le Y \sigma (tB+X)$ for a sufficiently small $t>0$ implies $A \le B$ if and only if $X$ is a positive scalar multiple of $Y$ or the associated operator monotone function $f$ with $\sigma$ has the form $f(t) = (at+b)/(ct+d)$, where $a,b,c,d$ are real numbers satisfying $ad-bc>0$.

In this paper we prove a lower bound for the determinant of the covariance matrix of quantum mechanical observables, which was conjectured by Gibilisco et al. and has the interpretation of uncertainty. The lower bound is given in terms of the commutator of the state and the observables and quantum Fisher information (generated by an operator monotone function).

We investigate the correspondence between operator monotone functions and Stieltjes functions. Then we shall prove that the class of nonnegative operator monotone functions is closed under certain operations, and applying those we decide the case when some binary operations become operator means.

A fixed point theorem and a norm inequality for operator means