The numerical range of a composition operator with conformal automorphism symbol

Rend. Circ. Mat. Palermo

2009

Let T be an operator on a Hilbert space H. The problem of computing of the norm of T , norm of selfcommutators of T , and the numerical radius of T are discussed in many papers and a number of textbooks. In this paper we determine the relationships between these values for self inverse operators and explain how we can determine any three of these ( T , [T * , T ] , {T * , T } , and the numerical radius of T ) by knowing any one of them. Also, we find the spectrum of T * T , [T * , T ] and {T * , T } in the case that T is self inverse and the spectrum of T * T is an interval. Finally, by giving some examples on automorphic composition operators, we show that these results make it possible to replace lengthy computation with quick ones.

Section: Examplesmentioning

confidence: 99%

“…Some of the earliest occurrences are [1,6,4] and [12] more recent treatments include [3,5,8,9,11,15,16] and [17].…”

mentioning

confidence: 99%

On the self-inverse operators

Rend. Circ. Mat. Palermo

2009

“…They also have shown that the numerical range of elliptic automorphism with order 2 is an ellipse with focus at ±1. In [1], the second author has completed their results by finding the exact value of the major axis of the ellipses. However, for the elliptic automorphisms with finite order k > 2, this is yet an open problem.…”

Section: Introductionmentioning

confidence: 99%

The numerical range of finite order elliptic automorphism composition operators

Heydari

Linear Algebra and its Applications

2015

Self Cite

“…They have a complete description of the spectrum and essential spectrum of [C * ϕ , C ϕ ], acting on the Hardy and Bergman spaces, when ϕ is an involution automorphism, i.e., when ϕ • ϕ = I. In this case, as an application of the results in the paper, they have determined the shape of the numerical range of C ϕ , which was determined in [1] by using matrix methods.…”

Section: Introductionmentioning

confidence: 99%

Self-commutators of automorphic composition operators on the Dirichlet space

2008

Proc. Amer. Math. Soc.

Self Cite

Abstract. Let ϕ be a conformal automorphism on the unit disk U and C ϕ : D −→ D be the composition operator on the Dirichlet space D induced by ϕ. In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators C * ϕ C ϕ , C ϕ C * ϕ and self-commutators of C ϕ , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.