On Eigenvalues and Boundary Curvature of the Numerical Rang of Composition Operators on Hardy Space ()
1. Introduction
For a bounded linear operator A on a Hilbert space
, the numerical range
is the image of the unit sphere of
under the quadratic form
associated with the operator. More precisely,
![](//html.scirp.org/file/2-5300804x18.png)
Thus the numerical range of an operator, like the spectrum, is a subset of the complex plane whose geometrical properties should say something about the operator.
One of the most fundamental properties of the numerical range is its convexity, stated by the famous Toeplitz-Hausdorff Theorem. Other important property of
is that its closure contains the spectrum of the operator,
is a connected set with a piecewise analytic boundary
[1] .
Hence, for all but finitely many points
, the radius of curvature
of
at p is well defined. By convention,
if p is a corner point of
, and
if p lies inside a flat portion of
.
Let
denote the distance from p to
, we define
the smallest constant such that
(1)
for all
with finite non-zero curvature.
By Donoghue’s theorem
whenever
. Therefore,
for all convexoid element A. Recall that convexoid element is an element such that its numerical range coincides with the convex hull of its spectrum. For non-convexoid A,
(2)
where the supremum in the right-hand side is taken along all points
with finite non-zero curvature.
The computation of
for arbitrary
matrix A is an interesting open problem. For
, we do not have an exact value of
. The question whether there exists a universal constant
, posed by Mathias [2] . Caston, et al. [3] prove the following inequalities:
(3)
Mirman a sequence of
Toeplitz nilpotent matrices
with
algrowing asymptotically as
is also found [3] . Hence, the answer to Mathias question is negative. However, the lower bound in (3) is still of some interest, at least for small values of n. The question of the exact rate of growth of
(it is
, or n, or something in between) remains open.
2. Composition Operator on Hardy Space
Let
denote the open unit disc in the complex plane, and the Hardy space H2 the functions ![]()
holomorphic in
such that
, with
denoting the n-th Taylor coefficient of f. The
inner product inducing the norm of
is given by
. The inner product of two functions f and g in
may also be computed by integration:
![]()
where
is positively oriented and f and g are defined a.e. on
via radial limits.
For each holomorphic self map
of
induces on
a composition operator
defined by the equation
. A consequence of a famous theorem of J. E. Littlewood [4] asserts that
is a bounded operator. (see also [5] [6] ).
In fact (see [6] )
![]()
In the case
, Joel H. Shapiro has been shown that the second inequality changes to equality if and only if
is an inner function.
A conformal automorphism is a univalent holomorphic mapping of
onto itself. Each such map is linear fractional, and can be represented as a product
, where
![]()
for some fixed
and
(See [7] ).
The map
interchanges the point p and the origin and it is a self-inverse automorphism of
.
Each conformal automorphism is a bijection map from the sphere
to itself with two fixed points (counting multiplicity). An automorphism is called:
elliptic if it has one fixed point in the disc and one outside the closed disc;
hyperbolic if it has two distinct fixed point on the boundary
, and
parabolic if there is one fixed point of multiplicity 2 on the boundary
.
For
, an r-dilation is a map of the form
. We call r the dilation parameter of
and in the case that
,
is called positive dilation. A conformal r-dilation is a map that is conformally conjugate to an
-dilation, i.e., a map
, where
and
is a conformal automorphism of
.
For
, an w-rotation is a map of the form
. We call w the rotation parameter of
. A straightforward calculation shows that every elliptic automorphism
of
must have the form
![]()
for some
and some
.
3. Main Results
In [8] , the shapes of the numerical range for composition operators induced on
by some conformal automorphisms of the unit disc specially parabolic and hyperbolic are investigated.
In [9] , V. Matache determined the shapes
in the case when the symbol of the composition operator the inducing functions are monomials or inner functions fixing 0. The numerical ranges of some compact composition operators are also presented.
Also, in [10] the spectrum of composition operators are investigated.
This facts will help in discussing and proving many of the results below.
Remark 3.1 If
,
, then
and
is a closed ellipticall disc whose boun-
dary is the ellipse of foci 0 and 1, having major/minor axis of length
and
. There- fore
.
Remark 3.2 If
,
, then
the closure of
. If w is the n-th root
of unity then
is the convex hull of all the n-th roots of unity and so
. If w is not a root of
unity the
is the union of
and the set
. In this case also
.
Remark 3.3 If
is hyperbolic with fixed point a,
, then
![]()
and
is a disc center at the origin. Therefore
where
is the numerical radius of
.
Remark 3.4 If
is parabolic, then
and
is a disc center at the origin. Therefore
.
Remark 3.5 If
is elliptic with rotation parameter w, and w is not a root of unity, then
and
is a disc center at the origin. Therefore
.
Therefore we have the following table for
.
Completing the Table
An elliptic automorphism
of
that does not fix the origin must have the form
, where
![]()
for some fixed
and
If we wish to show this dependence of
on p and w, we will denote the elliptic automorphism
by
.
If
is periodic then, surprisingly, the situation seems even murkier: For period 2 has been shown the closure of
is an elliptical disc with foci at
(Corollary 4.4. of [8] ). It is easy to see that
is open, also in [11] , the author completely determined
for period 2.
Theorem 3.6 If
is an elliptic automorphism with order 2 and P it’s only fixed point in open unit disc, then there is
such that
![]()
Proof. Let the operator A be self-inverse, i.e.,
but
, so
is an ellipse with foci at ±1 [12] . If
with
. Then
![]()
If
is an elliptic automorphism with order 2 and p it’s only fixed point in open unit disc, then ![]()
where
. Since
is a nontrivial self-inverse operator on Hardy space
and
is an inner
function, then
![]()
and so there is
such that:
![]()
But for period
then all we can say is that the numerical range of
has k-fold symmetry and we strongly suspect that in this case the closure is not a disc. Because the numerical range in this case is an open problem, so the completing of
is also open problem.
Acknowledgements
I thank the editor and the referee for their comments. Also, when the author is the responsible of establishing Center for Higher Education in Eghlid he is trying to write this paper, so I appreciate that center because of supporting me in conducting research.