Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators _Lα${\mathbf{L}}_{\alpha }$ in two variables, depending on the coupling parameter α?0$\alpha ?0$ that appears only in the boundary conditions. Our main concern is the spectral properties of _Lα${\mathbf{L}}_{\alpha }$, which turn out to be quite different for α<1$\alpha <1$ and for α>1$\alpha >1$. In particular, _Lα${\mathbf{L}}_{\alpha }$ has a unique self-adjoint realization for α<1$\alpha <1$ and many such realizations for α>1$\alpha >1$. In the more difficult case α>1$\alpha >1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.     

Symplectic Calabi–Yau 6-manifolds

In this article, we construct simply connected symplectic Calabi–Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T⁴${\mathbb{T}}^4$. Using our method, we also construct symplectic non-Kähler Calabi–Yau 6-manifolds with fundamental group Z$\mathbb{Z}$. This paper also produces the first examples of simply connected and non-simply connected symplectic Calabi–Yau 6-manifolds with fundamental groups _Zp×_Zq${\mathbb{Z}}_p\times {\mathbb{Z}}_q$, and Z×_Zq$\mathbb{Z}\times {\mathbb{Z}}_q$ for any p≥1$p\ge 1$ and q≥2$q\ge 2$via co-isotropic Luttinger surgery.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].