Analyticity of the Resolvent Function

Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.

Cesàro and Abel ergodic theorems for integrated semigroups

Let {S(t)}t ≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ 2 R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t 2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.

ϵ-CORRECTED SEIBERG–WITTEN PREPOTENTIAL OBTAINED FROM HALF-GENUS EXPANSION IN β-DEFORMED MATRIX MODEL

We consider the half-genus expansion of the resolvent function in the β-deformed matrix model with three-Penner potential under the AGT conjecture and the 0d–4d dictionary. The partition function of the model, after the specification of the paths, becomes the DF conformal block for fixed c and provides the Nekrasov partition function expanded both in [Formula: see text] and in ϵ = ϵ1+ϵ2. Exploiting the explicit expressions for the lower terms of the free energy extracted from the above expansion, we derive the first few ϵ corrections to the Seiberg–Witten prepotential in terms of the parameters of SU(2), Nf = 4, [Formula: see text] supersymmetric gauge theory.

Periodic solutions of Volterra integral equations

Consider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds, (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds. (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernelk. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent ofkis integrable in some sense. For a scalar convolution kernelksome explicit conditions are derived to determine whether or not the resolvent ofkis integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool.