. SP ] 1 6 Ja n 20 09 HIGHER ORDER SPECTRAL SHIFT , II . UNBOUNDED CASE

Representations of Hermitian Commutative ∗-algebras by Unbounded Operators

We give a spectral theorem for unital representions of Hermitian commutative unital ∗-algebras by possibly unbounded operators in a pre-Hilbert space. A more general result is known for the case in which the ∗-algebra is countably generated. 1. Statement of the Main Result Our main result is the following: Theorem 1. Let π be a unital representation of a Hermitian commutative unital ∗-algebra A...

ar X iv : m at h / 07 03 44 2 v 1 [ m at h . O A ] 1 4 M ar 2 00 7 OPERATOR INTEGRALS , SPECTRAL SHIFT AND SPECTRAL FLOW

We present a new and simple approach to the theory of multiple operator in-tegrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M. G. Kr...

An Optimal L-bound on the Krein Spectral Shift Function

and |ξA,B(λ)| ≤ n if A −B is rank n (2) are well known; see, for example, [5] or [6]. The Krein spectral shift function can also be defined for unbounded self-adjoint operators A,B and enjoys the same properties as long as their difference is trace class. The results of this paper extend to general unbounded operators A and B (as long as their difference is trace class) but for simplicity, we w...

Ja n 20 09 Kähler Ricci Flow on Fano Surfaces ( I )

We show the properties of the blowup limits of Kähler Ricci flow solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that Kähler Ricci flow converges to a Kähler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existence of Kähler Ricci soliton metrics on toric surfaces.

ar X iv : m at h / 06 01 74 3 v 1 [ m at h . SP ] 3 0 Ja n 20 06 DETERMINANTS OF ZEROTH ORDER OPERATORS