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Cancellability and regularity of operator connections with applications to nonlinear operator equations involving means
 Pattrawut Chansangiam^{1}Email author
https://doi.org/10.1186/s1366001509347
© Chansangiam 2015
 Received: 12 August 2015
 Accepted: 6 December 2015
 Published: 23 December 2015
Abstract
An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, and the transformer inequality. A normalized operator connection is called an operator mean. In this paper, we introduce and characterize the concepts of cancellability and regularity of operator connections with respect to operator monotone functions, Borel measures, and certain nonlinear operator equations. As applications, we investigate the existence and the uniqueness of solutions for operator equations involving various kind of operator means.
Keywords
 operator connection
 operator mean
 operator monotone function
MSC
 47A64
 47A62
 47A63
1 Introduction
 (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.

αweighted arithmetic means: \(A \,\triangledown_{\alpha }\, B = (1\alpha )A + \alpha B\);

αweighted geometric means: \(A \,\#_{\alpha }\, B = A^{1/2} ({A}^{1/2} B {A}^{1/2})^{\alpha } {A}^{1/2}\);

αweighted 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 \(f: \mathbb {R}^{+} \to \mathbb {R}^{+}\), \(f(x)=(x1)/\log{x}\), \(f(0) \equiv0\), and \(f(1) \equiv1\). Here, \(\mathbb {R}^{+}=[0, \infty)\).
 (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.
A connection σ on \(B(\mathbb {H})^{+}\) can be characterized via operator monotone functions as follows.
Theorem 1.1
([1])
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.
Theorem 1.2
([11])
The notion of monotone metrics arises naturally in quantum mechanics. A metric on a differentiable manifold of nbyn 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 onetoone 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.
Definition 2.1

left cancellable if for each \(A>0\), \(B \geqslant 0\), and \(C \geqslant 0\),$$\begin{aligned} A \,\sigma \,B = A \,\sigma \,C \quad\implies\quad B=C; \end{aligned}$$

right cancellable if for each \(A>0\), \(B \geqslant 0\), and \(C \geqslant 0\),$$\begin{aligned} B \,\sigma \,A = C \,\sigma \,A \quad\implies\quad B=C; \end{aligned}$$

cancellable if it is both left and right cancellable.
Lemma 2.2
Every nonconstant operator monotone function from \(\mathbb {R}^{+}\) to \(\mathbb {R}^{+}\) is injective.
Proof
Let \(f: \mathbb {R}^{+} \to \mathbb {R}^{+}\) be a nonconstant operator monotone function. Suppose there exist \(b>a \geqslant 0\) such that \(f(a)=f(b)\). Since f is monotone increasing (in the usual sense), \(f(x)=f(a)\) for all \(a \leqslant x \leqslant b\) and \(f(y) \geqslant f(b)\) for all \(y \geqslant b\). Since f is operator concave, f is concave in the usual sense and hence \(f(x)=f(b)\) for all \(x \geqslant b\). The case \(a=0\) contradicts the fact that f is nonconstant. For the case \(a>0\), suppose that there is a point \(c \in(0,a)\) such that \(0 \leqslant f(c)< f(a)\). The convexity of the function \(g(x)=xf(x)\) (see [1], Lemma 5.1) yields a contradiction. □
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.
Theorem 2.3
 (1)
σ is left cancellable;
 (2)
for each \(A \geqslant 0\) and \(B \geqslant 0\), \(I \,\sigma \,A = I \,\sigma \,B \implies A=B\);
 (3)
σ is not a scalar multiple of the lefttrivial mean;
 (4)f is injective, i.e., f is left cancellable in the sense that$$\begin{aligned} f \circ g = f \circ h \quad\implies\quad g=h ; \end{aligned}$$
 (5)
f is a nonconstant function;
 (6)
μ is not a scalar multiple of the Dirac measure \(\delta_{0}\) at 0.
Proof
(4) ⇒ (2): Assume that f is injective. Consider \(A \geqslant 0\) and \(B \geqslant 0\) such that \(I \,\sigma \,A = I \,\sigma \,B\). Then \(f(A) = f(B)\) by (1.3). Since \(f^{1} \circ f (x) =x\) for all \(x \in \mathbb {R}^{+}\), we have \(A=B\).
Theorem 2.4
 (1)
σ is right cancellable;
 (2)
for each \(A \geqslant 0\) and \(B \geqslant 0\), \(A \,\sigma \,I = B \,\sigma \,I \implies A=B\);
 (3)
σ is not a scalar multiple of the righttrivial mean;
 (4)
the transpose of f is injective;
 (5)
f is not a scalar multiple of the identity function \(x \mapsto x\);
 (6)
μ is not a scalar multiple of the Dirac measure \(\delta_{1}\) at 1.
Proof
Remark 2.5
The injectivity of the transpose of f does not imply the surjectivity of f. To see that, take \(f(x)=(1+x)/2\). Then the transpose of f is f itself.
The following results are characterizations of cancellability for connections.
Corollary 2.6
 (1)
σ is cancellable;
 (2)
σ is not a scalar multiple of the left/righttrivial mean;
 (3)
f and its transpose are injective;
 (4)
f is neither a constant function nor a scalar multiple of the identity function;
 (5)
μ is not a scalar multiple of \(\delta_{0}\) or \(\delta_{1}\).
Remark 2.7
The ‘order cancellability’ does not hold for general connections, even if we restrict them to the class of means. For each \(A,B>0\), it is not true that the condition \(I \,\sigma \,A \leqslant I \,\sigma \,B\) or the condition \(A \,\sigma \,I \leqslant B \,\sigma \,I\) implies \(A \leqslant B\). To see this, take σ to be the geometric mean. It is not true that \(A^{1/2} \leqslant B^{1/2}\) implies \(A \leqslant B\) in general.
3 Applications to certain nonlinear operator equations involving means
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.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.
Theorem 3.1
Proof
Similarly, we have the following theorem.
Theorem 3.2
Theorem 3.3
 (1)the operator equationhas a unique solution for any given \(A>0\) and \(B \geqslant 0\);$$\begin{aligned} A \,\sigma \,X = B \end{aligned}$$(3.1)
 (2)
f is unbounded and \(f(0)=0\);
 (3)f is surjective, i.e., f is right cancellable in the sense that$$\begin{aligned} g \circ f = h \circ f \quad\implies\quad g=h. \end{aligned}$$
Proof
(2) ⇒ (3): This follows directly from the intermediate value theorem.
(3) ⇒ (1): It is immediate from Theorem 3.1.
Now, let \(k>0\). The assumption (1) implies the existence of \(X \geqslant 0\) such that \(I \,\sigma \,X =kI\). Since \(f(X)=kI\), we have \(f(\lambda )=k\) for all \(\lambda \in \operatorname {Sp}(X)\). Since \(\operatorname {Sp}(X)\) is nonempty, there is \(\lambda \in \operatorname {Sp}(X)\) such that \(f(\lambda )=k\). Therefore, f is unbounded.
Example 3.4
The case \(0< p\leqslant 1\): The range of \(f_{p,\alpha }\) is the interval \([(1\alpha )^{1/p}, \infty)\). Hence, equation (3.2) is solvable if and only if \(\operatorname {Sp}(A^{1/2} B A^{1/2}) \subseteq[(1\alpha )^{1/p}, \infty)\), i.e., \(B \geqslant (1\alpha )^{1/p} A\).
The case \(1 \leqslant p< 0\): The range of \(f_{p,\alpha }\) is the interval \([0, (1\alpha )^{1/p})\). Hence, equation (3.2) is solvable if and only if \(\operatorname {Sp}(A^{1/2} B A^{1/2}) \subseteq[0, (1\alpha )^{1/p})\), i.e., \(B < (1\alpha )^{1/p} A\).
Example 3.5
Example 3.6
Example 3.7
The case \((1/3) < r < 0\): It is similar to the case \(0 < r < 1/3\). The operator equation (3.3) always has a unique solution.
4 Regularity of connections
In this section, we give various characterizations for the nonregularity of an operator connection.
Theorem 4.1
 (1)
\(f(0)=0\);
 (2)
\(\mu(\{0\}) = 0\);
 (3)
\(I \,\sigma \,0 = 0\);
 (4)
\(A \,\sigma \,0 =0\) for all \(A \geqslant 0\);
 (5)
for each \(A \geqslant 0\), the condition \(0 \in \operatorname {Sp}(A)\) implies \(0 \in \operatorname {Sp}(I \,\sigma \,A)\);
 (6)
for each \(A,X \geqslant 0\), the condition \(0 \in \operatorname {Sp}(A)\) implies \(0 \in \operatorname {Sp}(X \,\sigma \,A)\).
Proof
(5) ⇒ (1): We have \(0 \in \operatorname {Sp}(I \,\sigma \,0) = \operatorname {Sp}(f(0)I) = \{f(0)\}\), i.e. \(f(0)=0\).
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.
Remark 4.2
Theorem 4.3
 (1)
σ is nonregular;
 (2)
\(I \,\sigma \,P = P\) for each projection P.
Proof
(1) ⇒ (2): Assume that \(f(0)=0\) and consider a projection P. Since \(f(1)=1\), we have \(f(x)=x\) for all \(x \in\{0,1\} \supseteq \operatorname {Sp}(P)\). Thus \(I \,\sigma \,P = f(P)=P\).
(2) ⇒ (1): We have \(0=I \,\sigma \,0=f(0)I\), i.e. \(f(0)=0\). □
To prove the next result, recall the following lemma.
Lemma 4.4
([1])
 (1)
\(0< x<1 \implies x<f(x)<1\);
 (2)
\(1< x \implies1< f(x)< x\).
Theorem 4.5
 (1)
σ is nonregular;
 (2)
for each \(A \geqslant 0\), A is a projection if and only if \(I \,\sigma \,A = A\).
Proof
Since \(I \,\sigma \,A=A\), we have \(f(A)=A\) by (1.3). Hence \(f(x)=x\) for all \(x \in \operatorname {Sp}(A)\) by the injectivity of the continuous functional calculus. Since σ is a nontrivial mean, Lemma 4.4 implies that \(\operatorname {Sp}(A) \subseteq\{0,1\}\), i.e. A is a projection. □
Theorem 4.6
 (1)
σ is nonregular.
 (2)
The equation \(f(x)=0\) has a solution x.
 (3)
The equation \(f(x)=0\) has a unique solution x.
 (4)
The only solution to \(f(x)=0\) is \(x=0\).
 (5)
For each \(A>0\), the equation \(A \,\sigma \,X =0\) has a solution X.
 (6)
For each \(A>0\), the equation \(A \,\sigma \,X =0\) has a unique solution X.
 (7)
For each \(A>0\), the only solution to the equation \(A \,\sigma \,X =0\) is \(X=0\).
 (8)
The equation \(I \,\sigma \,X =0\) has a solution X.
 (9)
The equation \(I \,\sigma \,X =0\) has a unique solution X.
 (10)
The only solution to the equation \(I \,\sigma \,X =0\) is \(X=0\).
Proof
It is clear that (1) ⇒ (2), (6) ⇒ (5) and (10) ⇒ (8). Since f is injective by Theorem 2.3, we have (2) ⇒ (3) ⇒ (4).
(4) ⇒ (10): Let \(X \geqslant 0\) be such that \(I \,\sigma \,X =0\). Then \(f(X)=0\) by (1.3). By spectral mapping theorem, \(f(\operatorname {Sp}(X)) =\{0\}\). Hence, \(\operatorname {Sp}(X)=\{0\} \), i.e. \(X=0\).
(8) ⇒ (9): Consider \(X \geqslant 0\) such that \(I \,\sigma \,X =0\). Then \(f(X)=0\). Since f is injective with continuous inverse, we have \(X=f^{1}(0)\).
(9) ⇒ (6): Use congruence invariance.
(7) ⇒ (1): We have \(f(0)I = I \,\sigma \,0 =0\), i.e. \(f(0)=0\). □
Declarations
Acknowledgements
This research was supported by King Mongkut’s Institute of Technology Ladkrabang Research Fund grant no. KREF045710.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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