On Eigenvalues and Boundary Curvature of the Numerical Rang of Composition Operators on Hardy Space

For a bounded linear operator A on a Hilbert space  , let ( ) M A be the smallest possible constant in the inequality ( ) ( ) ( ) p p D A M A R A ≤ . Here, p is a point on the smooth portion of the boundary ( ) W A ∂ of the numerical range of A. ( ) p R A is the radius of curvature of ( ) W A ∂ at this point and ( ) p D A is the distance from p to the spectrum of A. In this paper, we compute the ( ) M A for composition operators on Hardy space H 2 .

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 ) ( A W is that its closure contains the spectrum of the op-erator, ) ( A W is a connected set with a piecewise analytic boundary ( ) . Hence, for all but finitely many points ( ) , we define ( ) for all ( ) with finite non-zero curvature.By Donoghue's theorem ( ) Recall that convexoid element is an element such that its numerical range coincides with the convex hull of its spectrum.For non-convexoid A, ( ) ( ) ( ) where the supremum in the right-hand side is taken along all points ( ) with finite non-zero curvature.The computation of ( ) Mirman a sequence of n n × Toeplitz nilpotent matrices n A with ( ) n M A algrowing asymptotically as log n 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 n M (it is log n , or n, or something in between) remains open.

Composition Operator on Hardy Space
Let  denote the open unit disc in the complex plane, and the Hardy space H 2 the functions ( ) ( ) f n denoting the n-th Taylor coefficient of f.The inner product inducing the norm of 2 H is given by ( ) ( ) . The inner product of two functions f and g in 2 H 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 ϕ of  induces on 2 H a composition operator C ϕ defined by the equ- ation ( ) . A consequence of a famous theorem of J. E. Littlewood [4] asserts that C ϕ is a bounded operator.(see also [5] [6]).
In fact (see [6]) In the case ( ) 0 0 ϕ ≠ , Joel H. Shapiro has been shown that the second inequality changes to equality if and only if ϕ 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 p w α ⋅ , where ( ) ( ) for some fixed p ∈  and w ∈ ∂ (See [7]).The map p α 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 r ∈  , an r-dilation is a map of the form ( ) r z rz δ = .We call r the dilation parameter of r δ and in the case that 0 r > , r δ is called positive dilation.A conformal r-dilation is a map that is conformally conjugate to an r -dilation, i.e., a map for some p ∈  and some w ∈ ∂ .

Main Results
In [8], the shapes of the numerical range for composition operators induced on 2 H by some conformal auto- morphisms of the unit disc specially parabolic and hyperbolic are investigated.

( )
W C ϕ 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.
dary is the ellipse of foci 0 and 1, having major/minor axis of length W C ϕ is a disc center at the origin.Therefore ( ) ( ) If ϕ is elliptic with rotation parameter w, and w is not a root of unity, then ( ) W C ϕ is a disc center at the origin.Therefore ( ) ( ) Therefore we have the following table for ( ) ϕ is an elliptic automorphism with rotation parameter is not a root of unity ( ) ϕ is an elliptic automorphism with rotation parameter is a root of unity ?

Completing the Table
An elliptic automorphism ϕ of  that does not fix the origin must have the form If ϕ is periodic then, surprisingly, the situation seems even murkier: For period 2 shown the closure of ( ) W C ϕ is an elliptical disc with foci at 1 ± (Corollary 4.4. of [8]).It is easy to see that ( ) [11], the author completely determined ( ) W C ϕ for period 2. Theorem 3.6 If ϕ is an elliptic automorphism with order 2 and P it's only fixed point in open unit disc, then there is 0 Proof.Let the operator A be self-inverse, i.e., 2  A I = but A I ≠ ± , so ( ) ( ) If ϕ is an elliptic automorphism with order 2 and p it's only fixed point in open unit disc, then But for period 2 k > then all we can say is that the numerical range of C ϕ 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 C ϕ is also open problem.
linear operator A on a Hilbert space  , the numerical range ( ) W A is the image of the unit sphere of  under the quadratic form , x Ax x → associated with the operator.More precisely, where r ∈  and α is conformal automorphism  .For w ∈ ∂ , an w-rotation is a map of the form w the rotation parameter of w ρ .A straightforward calculation shows that every elliptic automorphism ϕ of  must have the form ,  .If w is the n-th root of unity then ( ) W C ϕ is the convex hull of all the n-th roots of unity and so ( ) 0 M C ϕ = .If w is not a root of unity the ( ) W C ϕ is the union of  and the set { } : 0 n w n ≥ .In this case also If we wish to show this dependence of ϕ on p and w, we will denote the elliptic automorphism p