Search results for "Open unit"
showing 4 items of 4 documents
Perturbations of Jordan Blocks
2019
In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A0, introduced in Sect. 2.4: $$\displaystyle A_0=\begin {pmatrix}0 &1 &0 &0 &\ldots &0\\ 0 &0 &1 &0 &\ldots &0\\ 0 &0 &0 &1 &\ldots &0\\ . &. &. &. &\ldots &.\\ 0 &0 &0 &0 &\ldots &1\\ 0 &0 &0 &0 &\ldots &0 \end {pmatrix}: {\mathbf {C}}^N\to {\mathbf {C}}^N. $$ Zworski noticed that for every z ∈ D(0, 1), there are associated exponentially accurate quasimodes when N →∞. Hence the open unit disc is a region of spectral instability. We have spectral stability (a good resolvent estimate) in \(\mathbf {C}\setminus \overline {D(0,1)}\), since ∥A0∥ = 1. σ(A0) = {0}.
Cluster sets and quasiconformal mappings
2010
Certain classical results on cluster sets and boundary cluster sets of analytic functions, due to Iversen, Lindelof, Noshiro, Tsuji, Ohtsuka, Pommerenke, Carmona, Cufi and others, are extended to n-dimensional quasiconformal mappings. Unlike what is usually the case in the context of analytic functions, our considerations are not restricted to mappings of a disk or ball only. It is shown, for instance, that quasiconformal cluster sets and boundary cluster sets, taken at a non-isolated boundary point of an arbitrary domain, coincide. More refined versions are established in the special case where the domain is the open unit ball. These include cluster set considerations of the induced radial…
Nonexistence of global weak solutions for a nonlinear Schrodinger equation in an exterior domain
2020
We study the large-time behavior of solutions to the nonlinear exterior problem L u ( t , x ) = &kappa
Analytic structure in fibers of H∞(Bc0)
2020
Abstract Let H ∞ ( B c 0 ) be the algebra of all bounded holomorphic functions on the open unit ball of c 0 and M ( H ∞ ( B c 0 ) ) the spectrum of H ∞ ( B c 0 ) . We prove that for any point z in the closed unit ball of l ∞ there exists an analytic injection of the open ball B l ∞ into the fiber of z in M ( H ∞ ( B c 0 ) ) , which is an isometry from the Gleason metric of B l ∞ to the Gleason metric of M ( H ∞ ( B c 0 ) ) . We also show that, for some Banach spaces X, B l ∞ can be analytically injected into the fiber M z ( H ∞ ( B X ) ) for every point z ∈ B X .