1 Introduction
A general theory of connections and means for positive operators was given by Kubo and Ando [
1]. This theory is closely related to theory of operator inequalities and has deep applications in electrical network theory and mathematical physics. Let
\(B(\mathbb {H})\) be the algebra of bounded linear operators on a Hilbert space
\(\mathbb {H}\). The set of positive operators on
\(\mathbb {H}\) is denoted by
\(B(\mathbb {H})^{+}\). Denote the spectrum of an operator
X by
\(\operatorname {Sp}(X)\). For Hermitian operators
\(A,B \in B(\mathbb {H})\), the partial order
\(A \leqslant B\) indicates that
\(B-A \in B(\mathbb {H})^{+}\). The notation
\(A>0\) suggests that
A is a strictly positive operator. A
connection is a binary operation
σ on
\(B(\mathbb {H})^{+}\) such that for all positive operators
A,
B,
C,
D:
(M1)
monotonicity: \(A \leqslant C, B \leqslant D \implies A \,\sigma \,B \leqslant C \,\sigma \,D\);
(M2)
transformer inequality: \(C(A \,\sigma \,B)C \leqslant (CAC) \,\sigma \,(CBC)\);
(M3)
continuity from above: for \(A_{n},B_{n} \in B(\mathbb {H})^{+}\), if \(A_{n} \downarrow A\) and \(B_{n} \downarrow B\), then \(A_{n} \,\sigma \,B_{n} \downarrow A \,\sigma \,B\). Here, \(A_{n} \downarrow A\) indicates that \((A_{n})\) is a decreasing sequence converging strongly to A.
Two trivial examples are the left-trivial mean
\(\omega_{l} : (A,B) \mapsto A\) and the right-trivial mean
\(\omega_{r}: (A,B) \mapsto B\). The sum
\((A,B) \mapsto A+B\) is clearly a connection. A connection was modeled from the notion of parallel sum, introduced by Anderson and Duffin [
2],
$$\begin{aligned} A : B = \bigl(A^{-1}+B^{-1}\bigr)^{-1},\quad A,B>0. \end{aligned}$$
This notion plays an important role in the analysis of electrical networks.
From the transformer inequality, every connection is invariant consider congruences in the sense that for each
\(A,B \geqslant 0\) and
\(C>0\) we have
$$\begin{aligned} C(A \,\sigma \,B)C = (CAC) \,\sigma \,(CBC). \end{aligned}$$
A
mean is a connection
σ with normalized condition
\(I \,\sigma \,I = I\) or, equivalently, fixed-point property
\(A \,\sigma \,A =A\) for all
\(A \geqslant 0\). The class of Kubo-Ando means cover many well-known operator means in practice,
e.g.
-
α-weighted arithmetic means: \(A \,\triangledown_{\alpha }\, B = (1-\alpha )A + \alpha B\);
-
α-weighted geometric means: \(A \,\#_{\alpha }\, B = A^{1/2} ({A}^{-1/2} B {A}^{-1/2})^{\alpha } {A}^{1/2}\);
-
α-weighted harmonic means: \(A \,!_{\alpha }\, B = [(1-\alpha )A^{-1} + \alpha B^{-1}]^{-1}\);
-
logarithmic mean: \((A,B) \mapsto A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}\) where \(f: \mathbb {R}^{+} \to \mathbb {R}^{+}\), \(f(x)=(x-1)/\log{x}\), \(f(0) \equiv0\), and \(f(1) \equiv1\). Here, \(\mathbb {R}^{+}=[0, \infty)\).
See [
3,
4], [
5], Section 3, and [
6], Chapter 5.
It is a fundamental that there are one-to-one correspondences between the following objects:
(1)
operator connections on \(B(\mathbb {H})^{+}\);
(2)
operator monotone functions from \(\mathbb {R}^{+}\) to \(\mathbb {R}^{+}\);
(3)
finite (positive) Borel measures on \([0,1]\);
(4)
monotone (Riemannian) metrics on the smooth manifold of positive definite matrices.
Recall that a function
\(f: \mathbb {R}^{+} \to \mathbb {R}^{+}\) is said to be
operator monotone if
$$\begin{aligned} A \leqslant B \quad\implies\quad f(A) \leqslant f(B) \end{aligned}$$
for all positive operators
\(A,B \in B(\mathbb {H})\) and for all Hilbert spaces
\(\mathbb {H}\). This concept was introduced in [
7]; see also [
8], Chapter V, [
5], Section 2, and [
6], Chapter 4. Concrete examples of operator monotone functions are provided in [
9]. A remarkable fact is that (see [
10]) a function
\(f: \mathbb {R}^{+} \to \mathbb {R}^{+}\) is operator monotone if and only if it is
operator concave,
i.e.
$$\begin{aligned} f\bigl((1-\alpha )A + \alpha B\bigr) \geqslant (1-\alpha ) f(A) + \alpha f(B), \quad \alpha \in(0,1), \end{aligned}$$
for all positive operators
\(A,B \in B(\mathbb {H})\) and for all Hilbert spaces
\(\mathbb {H}\).
A connection σ on \(B(\mathbb {H})^{+}\) can be characterized via operator monotone functions as follows.
We call f the representing function of σ. A connection also has a canonical characterization with respect to a Borel measure via a meaningful integral representation as follows.
We call
μ the
associated measure of
σ. A connection is a mean if and only if
\(f(1)=1\) or its associated measure is a probability measure. Hence every mean can be regarded as an average of weighted harmonic means. From (
1.1) and (
1.2),
σ and
f are related by
$$ f(A) = I \,\sigma \,A,\quad A \geqslant 0. $$
(1.3)
A connection
σ is said to be
symmetric if
\(A \,\sigma \,B = B \,\sigma \,A\) for all
\(A,B \geqslant 0\).
The notion of monotone metrics arises naturally in quantum mechanics. A metric on a differentiable manifold of
n-by-
n positive definite matrices is a continuous family of positive definite sesquilinear forms assigned to each invertible density matrix in the manifold. A monotone metric is a metric with the contraction property under stochastic maps. It was shown in [
12] that there is a one-to-one correspondence between operator connections and monotone metrics. Moreover, symmetric metrics correspond to symmetric means. In [
13], the author defined a symmetric metric to be
nonregular if
\(f(0)=0\) where
f is the associated operator monotone function. In [
14],
f is said to be
nonregular if
\(f(0) = 0\), otherwise
f is
regular. It turns out that the regularity of the associated operator monotone function guarantees the extendability of this metric to the complex projective space generated by the pure states (see [
15]).
In the present paper, we introduce the concept of cancellability for operator connections in a natural way. Various characterizations of cancellability with respect to operator monotone functions, Borel measures, and certain operator equations are provided. It is shown that a connection is cancellable if and only if it is not a scalar multiple of trivial means. Applications of this concept go to certain nonlinear operator equations involving operator means. It is shown that such equations are always solvable if and only if f is unbounded and \(f(0)=0\) where f is the associated operator monotone function. We also characterize the condition \(f(0)=0\) for arbitrary connections without assuming the symmetry. Such a connection is said to be nonregular.
This paper is organized as follows. In Section
2, the concept of cancellability of operator connections is defined and characterized. Applications of cancellability to certain nonlinear operator equations involving operators means are explained in Section
3. We investigate the regularity of operator connections in Section
4.
2 Cancellability of connections
The concept of cancellability for scalar means was considered in [
16]. We generalize this concept to operator means or, more generally, operator connections as follows.
A similar result for this lemma under the restriction that
\(f(0)=0\) was obtained in [
17]. The left cancellability of connections is now characterized as follows.
Recall that the
transpose of a connection
σ is the connection
$$ (A,B) \mapsto B \,\sigma \,A. $$
If
f is the representing function of
σ, then the representing function of the transpose of
σ is given by the
transpose of
f (see [
1], Corollary 4.2), defined by
$$ x \mapsto xf(1/x),\quad x>0. $$
A connection is
symmetric if it coincides with its transpose.
The following results are characterizations of cancellability for connections.
3 Applications to certain nonlinear operator equations involving means
Cancellability of connections can be restated in terms of the uniqueness of certain operator equations as follows. A connection
σ is left cancellable if and only if
for each given \(A>0\) and \(B \geqslant 0\), if the equation \(A \,\sigma \,X =B\) has a solution X, then it has a unique solution.
The similar statement for right cancellability holds. In this section, we characterize the existence and the uniqueness of a solution of the operator equation
\(A \,\sigma \,X =B\). The equations of this type with specific operator means
σ are also considered.
Similarly, we have the following theorem.
Next, we investigate certain nonlinear operator equations involving operator means. First, consider a class of parametrized means, namely, the
quasi-arithmetic power mean
\(\#_{p,\alpha }\) with exponent
\(p \in[-1,1]\) and weight
\(\alpha \in(0,1)\), defined by
$$ A \,\#_{p,\alpha }\, B = \bigl[(1-\alpha )A^{p} + \alpha B^{p} \bigr]^{1/p}. $$
Its representing function of this mean is given by
$$ f_{p,\alpha }(x) = \bigl(1-\alpha +\alpha x^{p}\bigr)^{1/p}. $$
The special cases
\(p=1\) and
\(p=-1\) are the
α-weighted arithmetic mean and the
α-weighted harmonic mean, respectively. The case
\(p=0\) is defined by continuity and, in fact,
\(\#_{0,\alpha } = \# _{\alpha }\) and
\(f_{0,\alpha }(x)=x^{\alpha }\).
Next, we will consider a parametrized symmetric mean. For each
\(r \in[-1,1]\), recall that the function
$$\begin{aligned} g_{r} (x) = \biggl(\frac{3r-1}{3r+1} \biggr) \frac{x^{\frac {3r+1}{2}} -1}{x^{\frac{3r-1}{2}} -1},\quad x \geqslant 0, \end{aligned}$$
is operator monotone (see [
4]). This function satisfies
\(g_{r}(1)=1\) and
\(g_{r}(x) = x g_{r}(1/x)\). Thus it associates to a unique symmetric operator mean, denoted by
\(\diamondsuit_{r}\). In particular,
$$\begin{aligned} \diamondsuit_{1} = \triangledown,\qquad \diamondsuit_{0} = \#,\qquad \diamondsuit_{-1} = !. \end{aligned}$$
The operator means
\(\diamondsuit_{1/3}\) and
\(\diamondsuit_{-1/3}\) are the logarithmic mean and its dual.
4 Regularity of connections
In this section, we give various characterizations for the non-regularity of an operator connection.
We say that a connection
σ is
nonregular if one of the conditions in Theorem
4.1 holds (and thus they all do), otherwise
σ is
regular. Hence, regular connections correspond to regular operator monotone functions and regular monotone metrics.
To prove the next result, recall the following lemma.
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Competing interests
The author declares that he has no competing interests.