Differences Of Composition Operators On Bloch Type Spaces

Let D be the open unit disk in the complex plane C, H(D) be the set ofall analytic functions on D and S(D) the set of self-analytic functions on D.For a givenφ∈S(D), it can induce a composition operator C_φdefined byC_φf = f (?)φ, f∈H(D).For 0 <α<∞, we say that a function f with f∈H(D) is in (?)^αif itsatisfiesMoreover, we say that it is in (?)₀^αif it also satisfiesIn this thesis we first study the boundedness and compactness of di?erenceof composition operators between (?)^α((?)₀^α)and Bβ((?)₀^β) and give their su?cientand necessary conditions. Then we study the topological structure of the setof composition operators on Bloch type space C((?)^α)(C((?)₀^α)) using the com-pactness of the di?erence of composition operators. We know that the set ofcompact composition operators on (?)₀^αis path connected and give the su?-cient condition of any two composition operators C_φand C_ψto be in the samecomponent on C((?)^α)(C((?)₀^α)).