A note on the spectral theorem

We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)^–1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.This research was supported in part by Department of Energy Grant No. DE-AS05-80ER10711 and National Science Foundation Grant No. DMA-8312451.     

A note on Weyl's theorem

The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder's theorem to hold.

     

A note on Rouche's theorem

The purpose of this note is to show a characterization of Rouche's result concerning the zeros of pairs of analytic functions, as well as its geometric interpretation.     

Interphase Hamiltonian and first-order phase transitions: A generalization of the Lee-Yang theorem

We generalize the Pirogov-Sinai theory and prove the results applicable to first-order phase transitions in the case of both bulk and surface phase lattice models. The region of first-order phase transitions is extended with respect to the chemical activities to the entire complex space ℂ^Ф, where Φ is the set of phases in the model. We prove a generalization of the Lee-Yang theorem: as functions of the activities, the partition functions with a stable boundary condition have no zeros in ℂ^Ф. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 98–123, October, 2007.     

A note on the theorem of Leray-Schauder

Using elemental methods of Topology, theory of degree and theory of metric continua, we prove a new version of the theorem of Leray-Schauder. It provides the existence of arc-connected set of solutions. This result may have a lot of applications in a large variety of problems. Although the assumption of the theorem is not easy to verify in practice, this theorem could be an important tool to prove not only the existence of set of solutions but also the existence of a set of solution which is homeomorphic to the interval [0,1]⊂R.     

A note on generalized Weyl's theorem

We prove that if either T or T^∗ has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every f∈H(σ(T)). An application is given for algebraically paranormal operators.     

A note on Laplace's expansion theorem

A short proof of Laplace's expansion theorem is given. The proof is elementary and can be presented at any level of undergraduate studies where determinants are taught. It is derived directly from the definition so that the theorem may be used as a starting point for further investigation of determinants.