Some inequalities of operator monotone functions

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

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.

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

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.

Loewner's theorem in several variables

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.

Uniqueness of the maximal extension of a monotone operator

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 $ad-bc>0$.

The mixed boundary value problem has been treated recently by several authors [4; 5; 6; 7]. In this note we give an existence proof using subharmonic functions. Consider a 2-dimensional multiply connected domain D with the boundary r, which consists of k closed curves ri, F2, * * *, rk. The boundary curves IX (i =1, 2, * * , k) are assumed to have continuous tangents. A real-valued continuous function f(r) is defined on a part a of the boundary. (The boundary points are denoted by t.) a consists of a finite number of disjoint closed curves or arcs ai. If ai is an arc, then we denote by P', P' its two end points. If(<) | < M. The remaining part of r we denote by ,. Another real valued continuous function g(?) is defined on the closure of ,3. The problem is to determine a function u(z) in D, such that: (a) u(z) is harmonic in D, (b) limz_r u(z) =f(r) on ao and lim supzr I u(z) | < Mat =P', P , (c) limz_r u(z) is continuous for r on , (d) cu/cn=g(v) for r on 3, where the differentiation is along the inward normal n. We shall treat first the problem by assuming g(v) =0.2 Since Neumann's problem can be solved for smooth boundary, the assumption g( )=0 on ( is not a restriction. Define the class 7 of admissible functions v(z) with the properties: (a) v(z) is continuous subharmonic in D, (b) lim supz_r v(z) <f(r) for r on a, (c) lim24 v(z) exists and is continuous for v on A, (d) lim inf Av/An ?0 on (. The lim inf is taken along the normal pointing into the interior of D.

We propose a variable metric extension of the forward–backward-forward algorithm for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. In turn, this framework provides a variable metric splitting algorithm for solving monotone inclusions involving sums of composite operators. Monotone operator splitting methods recently proposed in the literature are recovered as special cases.

Finding a zero of the sum of two monotone operators is one of the most important problems in monotone operator theory, and the forward-backward algorithm is the most prominent approach for solving this type of problem. The aim of this paper is to present a new preconditioning forward-backward algorithm to obtain the zero of the sum of two operators in which one is maximal monoton and the other one is M-cocoercive, where M is a linear bounded operator. Furthermore, the strong convergence of the proposed algorithm, which is a broader variant of previously known algorithms, has been proven in Hilbert spaces. We also use our algorithm to tackle the convex minimization problem and show that it outperforms existing algorithms. Finally, we discuss several image restoration applications.

We present two algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive operator in Hilbert spaces. We show that these two algorithms converge strongly to the minimum norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive operator.

We propose adjointations between operator orderings, which convert any operator inequalities/identities associated with certain binary operations to new ones. Then we prove that a continuous function \(f:(0,\infty) \to (0,\infty)\) is operator monotone increasing if and only if \(f(A \: !_t \: B) \leq f(A) \: !_t \: f(B)\) for any positive operators \(A,B\) and scalar \(t \in [0,1]\). Here, \(!_t\) denotes the \(t\)-weighted harmonic mean. As a counterpart, \(f\) is operator monotone decreasing if and only if the reverse of preceding inequality holds. Moreover, we obtain many characterizations of operator monotone increasingness/decreasingness in terms of operator means. These characterizations lead to many operator inequalities involving means.

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

On [formula omitted]-factorizable topological groups