AbstractA bounded linear operator A is said to be quadratic if there is a polynomial p of degree 2 suchthat p(A)=0. Square zero operators, involutions, and idempotents are all typical quadraticoperators. We will give characterizations of matrices could be expressed as commutators of twosquare zero matrices, and explain some related results about limits of commutators of two squarezero operators...
AbstractIt would be useful to deal with complex geometry when examining quantum gravity as in the case of no-boundary proposal by Hartle and Hawking. However, there would be too many saddles for complexified gravity, and it is necessary to determine which are allowable geometries in the sense of Witten. We consider three-ddimensional gravity theory with positive consmological constant described...