Some properties of sums of weighted composition operators on the Fock space

Let H be a Hilbert space. For each f∈H, we define a multiplication operator M_φ by M_φ (f)=φf. Let φ be an entire function. For each f belongs to the Fock space F^2, the composition operator C_φ is defined by C_φ (f)=f∘φ. For entire functions ψ, φ and f∈F^2, the weighted composition operator C_(ψ,φ) on F^2 are given by C_(ψ,φ) (f)=ψ.(f∘φ). Let T be a bounded operator on H, the set W(T)={⟨Tf,f⟩:‖f‖=1} is called the numerical range of T. In this paper, we find the point spectrum of some operators C_(ψ_1,φ_1 )+C_(ψ_2,φ_2 ), when φ_1 and φ_2 have the some fixed point. Moreover, we obtain an invariant subspace for the operator (C_(ψ_1,φ_1 )+C_(ψ_2,φ_2 ) )^*. Then by these results, for compact operators C_(ψ_1,φ_1 ) and C_(ψ_2,φ_2 ), we find the spectrum of C_(ψ_1,φ_1 )+C_(ψ_2,φ_2 ). Then for φ_1 and φ_2 which have the some fixed point, we investigate the numerical range of C_(ψ_1,φ_1 )+C_(ψ_2,φ_2 ).

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