Hyperbolic monotonicity in the Hilbert ball
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We first characterize ρ-monotone mappings on the Hilbert ball by using their resolvents and then study the asymptotic behavior of compositions and convex combinations of these resolvents. Copyright © 2006 E. Kopeck´a and S. Reich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Monotone operator theory has been intensively developed with many applications to Convex and Nonlinear Analysis, Partial Differential Equations, and Optimization. In this note we intend to apply the concept of (hyperbolic) monotonicity to Complex Analysis. As we will see, this application involves the generation theory of one-parameter continuous semigroups of holomorphic mappings. Let (H, ·, ·) be a complex Hilbert space with inner product ·, · and norm | · |, and let B := {x ∈ H : |x| < 1} be its open unit ball. The hyperbolic metric ρ on B × B [5, page 98] is defined by
ρ(x, y) := argtanh 1 − σ(x, y)
1/2
,
(1.1)
x, y ∈ B.
(1.2)
where
1 − |x|2 1 − | y |2
σ(x, y) :=
1 − x, y 2
,
A mapping g : B → B is said to be ρ-nonexpansive if
ρ g(x),g(y) ≤ ρ(x, y)
(1.3)
for all x, y ∈ B. It is known (see, for instance, [5, page 91]) that each holomorphic selfmapping of B is ρ-nonexpansive. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 78104, Pages 1–15 DOI 10.1155/FPTA/2006/78104
2
Hyperbolic monotonicity in the Hilbert ball
Recall that if C is a subset of H, then a (single-valued) mapping f : C → H is said to be monotone if
Re x − y, f (x) − f (y) ≥ 0,
x, y ∈ C.
(1.4)
Equivalently, f is monotone if
Re x, f (x) + y, f (y)
≥ Re
y, f (x) + x, f (y) ,
x, y ∈ C.
(1.5)
It is also not difficult to see that f is monotone if and only if |x − y | ≤ x + r f (x) − y + r f (y) ,
x, y ∈ C,
(1.6)
for all (small enough) r > 0. Let I denote the identity operator. A mapping f : C → H is said to satisfy the range condition if (I + r f )(C) ⊃ C,
r > 0.
(1.7)
If f is monotone and satisfies the range condition, then the mapping Jr : C → C, welldefined for positive r by Jr := (I + r f )−1 , is called a (nonlinear) resolvent of f . It is clearly nonexpansive, that is, 1-Lipschitz: Jr x − Jr y ≤ |x − y |,
x, y ∈ C.
(1.8)
As a matter of fact, this resolvent is even firmly nonexpansive: Jr x − Jr y ≤ Jr x − Jr y + s x − Jr x − y − Jr y
(1.9)
for all x and y in C and for all positive s. This is a direct consequence of (1.6) because x − Jr x = r f (Jr x) and y − Jr y = r f (Jr y) for all x and y in C. We remark in passing that, conversely, each firmly nonexpansive mapping is a resolvent of a (possibly set-valued) monotone operator. To see this, let T : C → C be firmly nonexpansive. Then the operator M :=
[Tx,x − Tx] : x ∈ C
(1.10)
is monotone because T satisfies (1.9). We now turn to the concept of hyperbolic monotoni
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