Eigenvalue estimates for Dirac operators coupled to instantons

We prove a sharp lower bound for the first positive eigenvalue of Dirac operators coupled to instantons and discuss the limit case.     

Factorisation of positive definite operators

In this paper we prove Reade’s result for the positive definite C ¹ kernels by using the factorisation method used by Kühn. Received: 10 April 2007, Revised: 3 December 2007     

On extensions of positive definite integral Fredholm operators

The main result of this paper states that a positive definite Fredholm integral operator acting on L²([0,1]) can be modified on a Lebesque measurable set DmitDelta in [0,1]² such that the resulting operator is positive definite and its resolvent kernel is zero on DmitDelta . This answers a question raised in [3]. The proof is based on extension results for positive definite operator matrices and their connection to generalized determinants.     

Dissolving four-manifolds and positive scalar curvature

We prove that many simply connected symplectic four-manifolds dissolve after connected sum with only one copy of S ² ×S ² . For any finite group G that acts freely on the three-sphere we construct closed smooth four-manifolds with fundamental group G which do not admit metrics of positive scalar curvature, but whose universal covers do admit such metrics. Mathematics Subject Classification (2000):57R57, 53C25, 57R55.We are very grateful to R. Gompf for valuable comments. The first and the second authors are members of the European Differential Geometry Endeavour (EDGE), Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme.     

On symmetrizable operators of which some iteration satisfies a positive definite condition

Considered are linear (in general, unbounded) operators A, defined on a set R which is dense in the Hubert Space X, which are symmetrizable by a symmetric operator H in R. Under the condition that there exists an integer p 0 for which (HA^P _x, x) 0 for any x R, the spectral properties of the operator A and the solutions of the equation x - Ax=y, x, y R are investigated. The results obtained are applied to investigating some boundary-value problems for differential equations.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 71–76, January, 1969.     

Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators

LetA andB be two anticommuting self-adjoint operators andV() be a symmetric operator in a Hilbert space, where >0 is a parameter. It is proven that, under some conditions forV(), the resolvents of A+² B±²|B|+V() converge as . Applications to the nonrelativistic-limit problem of Dirac operators and supersymmetry are discussed.This work is supported by the Grant-In-Aid 0560139 for science research from the Ministry of Education, Japan.     

Operator-valued positive definite kernels on tubes

In this paper we generalize the concept of an infinite positive measure on a -algebra to a vector valued setting, where we consider measures with values in the compactification of a convex coneC which can be described as the set of monoid homomorphisms of the dual coneC ^* into [0, ]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operator valued positive definite functions on locally compact abelian groups and likewise to generalizations of Nussbaum's Theorem on positive definite functions on cones. In the latter case we use the Laplace transform to realize the corresponding Hilbert spaces by holomorphic functions on tube domains.     

Extrapolated positive definite and positive semi-definite splitting methods for solving non-Hermitian positive definite linear systems

Applications of Mathematics - Recently, Na Huang and Changfeng Ma in (2016) proposed two kinds of typical practical choices of the PPS method. In this paper, we extrapolate two versions of the PPS...     

Weinstein positive definite functions

Positivity - In this paper, we introduce the notion of the Weinstein positive definite functions and we state a version of Bochner’s theorem. Furthermore, we study the strictly Weinstein...     

On positive definite measures

In this paper we study the Fourier transform of unbounded measures on a locally compact groupG. After a short introductory section containing background material, especially results established byL. Argabright andJ. Gil De Lamadrid we turn to the main subjects of the paper: first we characterize \(\Re \left( G \right), \mathfrak{J}\left( G \right)\) andB(G) cones in \(\mathfrak{W}\left( G \right)\) . After that we establish the subspace \(\mathfrak{W}_\Delta \left( G \right)\) of \(\mathfrak{W}\left( G \right)\) which contains \(\mathfrak{W}_p \left( G \right)\) , the linear span of all positive definite measures.     

Remarks on the spectrum of Dirac operators

This paper concerns the number of eigenvalues of the Dirac operator in ]-1,1[ and gives the number of linear independent eigenfunctions with total angular momentum k in [, 1[.     

Symplectic Dirac operators on Hermitian symmetric spaces

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.