Some operator monotone functions

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].

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.

Loewner's theorem in several variables

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

Let $A, B$ be positive operators and let f be any operator monotone function. We obtain inequalities for $|||f(A)X - Xf(B)|||$ in terms of $|||f\left(|AX - XB| \right) |||$ for every unitarily invariant norm. The case $X = I$ was considered by T. Ando [Math. Z., 197 (1988), pp. 403--409], and some of our results reduce to his results in this special case. Some related inequalities are obtained.

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

Let T be a positive linear operator on L1 of a probability space such that II T1j i: 1. In this note we consider the following question: Under what condition is T multiplicative on Let (X, F, m) be a probability space and L1=L1(X, F, m) the Banach space of equivalence classes of integrable complex-valued functions on X. Let T be a positive linear operator on L1 such that II TIIl< 1, and define G(T)={f E L1; If I = I Tf I = 1). The main purpose of this note is to prove the following: THEOREM. If G(T) is total in L1, i.e., the linear manifold generated by G(T) is dense in L1 in the norm topology, then for any bounded functions f and g we have T(fg)= (Tf)(Tg). As a corollary of the Theorem we have the following result; essentially the same idea has been used by Halmos [2, p. 45] to find a necessary and sufficient condition for a unitary operator on L2=L2(X, Y, m) to be induced by an automorphism of the measure algebra (F(m), m) associated with (X, F, m). COROLLARY. A necessary and sufficient condition that T be induced by a homomorphism of the measure algebra (f(m), m) into itself is that G(T) be total in L1. For the proof of the theorem we need some lemmas. LEMMA 1. Let K be a compact Hausdorff space and C(K) the Banach algebra of all complex-valued continuous functions f on K wvith norm lIf IIU=sup{If(s)I; s e K}. Let U be a positive linear operator on C(K) such that U1=1. Let f,geC(K), IfI=lgi=1, and IUfI=IUgi=1. Then U(fg)=(Uf)(Ug)Received by the editors February 28, 1973 and, in revised form, May 10, 1973. AMS (MOS) subject classifications (1970). Primary 47B55, 47A35; Secondary 28A65, 46J10.

SOME SPECIAL METHODS OF HOMEOMORPHISM THEORY IN INFINITE-DIMENSIONAL TOPOLOGY

A real-valued continuous function f(2) defined on a (finite or infinite) interval of the real line is said to be operator monotone if, for any pair of bounded selfadjoint operators A, B on an infinite dimensional Hilbert space, A > B implies f (A) > f(B). Here f(A) and f(B) are defined according to the usual functional calculus, and > refers to the order relation induced by the cone of positive (semi-definite) operators. Operator monotony is quite a restrictive condition. In fact, for p > 0 the function f(2): = 2 p is operator monotone on the half-line [0, c~) if and only if 0 1 the function f(2) = 2 p possesses some property akin to operator monotony. In this direction Furuta I-3] recently observed that if r > 0, p > 0, q > 1 satisfies (1 + 2r)q > p + 2r then A > B > 0 implies A tp § 2,)/q > (ArBPA r) l/~. If p = 2r and q = 2, the requirement is automatically satisfied, hence A p > (Ap/2BPAP/2)I/2 whenever A > B > 0. When A is invertible, this means that

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.

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).