Commutators with finite spectrum
The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum σ([dx,d′x]) is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d,d′] is the inner derivation implemented by a.
Identifying Derivations Through the Spectra 2 . Commutators with Symmetric Spectra
We consider the relationship between derivations d and g of a Banach algebra B that satisfy σ(g(x)) ⊆ σ(d(x)) for every x ∈ B, where σ( . ) stands for the spectrum. It turns out that in some basic situations, say if B = B(X), the only possibilities are that g = d, g = 0, and, if d is an inner derivation implemented by an algebraic element of degree 2, also g = −d. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general C∗-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition ‖[b, x]‖ ≤ M‖[a, x]‖ for all selfadjoint elements x from a C∗-algebra B, where a, b ∈ B and a is normal. Mathematics Subject Classification (2010). 46H05, 46L57, 47B47.
Rapid Spatially Coupled AVO Inversion
The Bayesian methodology yields a natural framework for many geophysical inverse problem, see e.g. (Tarantola and Valette, 1982; Tarantola, 1987; Duijndam, 1988a; Duijndam, 1988b). The solution of a Bayesian inverse problem is represented by the posterior distribution. From the posterior distribution the best estimate and the corresponding uncertainty can be extracted and a set of plausible solutions can be drawn.