Quasi-similarity of contractions having a 2×1 characteristic function.
We propose a noncommutative holomorphic functional calculus on absolutely convex domains for a Banach algebra homomorphism π of a finite-dimensional solvable Lie algebra 𝔤 in terms of quasispectra σ(π). In particular, we prove that the joint spectral radius of a compact subset in a solvable operator Lie subalgebra coincides with the geometric spectral radius with respect to a quasispectrum.
We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak* analogue. We introduce and study analogues of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L₁[0,1] by an ℓ₁-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.
Here we present and example and some results suggesting that there is no infinite-dimensional reflexive subspace Z of L1 ≡ L1[0,1] such that the quotient L1/Z is isomorphic to a subspace of L1.