Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 convergence without any...

Given a completely bounded map $u:Z\to M$ from an operator space $Z$ into a von Neumann algebra (or merely a unital dual algebra) $M$, we define $u$ to be $C$-semidiscrete if for any operator algebra $A$, the tensor operator $I_A\otimes u$ is bounded from $A{\otimes }_{\min }Z$ into $A{\otimes }_{\mathrm{nor}}M$, with norm less than $C$. We investigate this property and characterize it by suitable approximation properties, thus generalizing the Choi-Effros characterization of semidiscrete von Neumann algebras. Our work is an extension of some recent work of Pisier on an analogous...

We study some algebraic properties of commutators of Toeplitz operators on the Hardy space of the bidisk. First, for two symbols where one is arbitrary and the other is (co-)analytic with respect to one fixed variable, we show that there is no nontrivial finite rank commutator. Also, for two symbols with separated variables, we prove that there is no nontrivial finite rank commutator or compact commutator in certain cases.

In this paper we investigate finite rank operators in the Jacobson radical ${ℛ}_{𝒩\otimes ℳ}$ of $\mathrm{A}lg(𝒩\otimes ℳ)$, where $𝒩$, $ℳ$ are nests. Based on the concrete characterizations of rank one operators in $\mathrm{A}lg(𝒩\otimes ℳ)$ and ${ℛ}_{𝒩\otimes ℳ}$, we obtain that each finite rank operator in ${ℛ}_{𝒩\otimes ℳ}$ can be written as a finite sum of rank one operators in ${ℛ}_{𝒩\otimes ℳ}$ and the weak closure of ${ℛ}_{𝒩\otimes ℳ}$ equals $\mathrm{A}lg\left(𝒩\otimes ℳ\right)$ if and only if at least one of $𝒩$, $ℳ$ is continuous.

We describe the C*-algebra associated with the finite sections discretization of truncated Toeplitz operators on the model space K2u where u is an infinite Blaschke product. As consequences, we get a stability criterion for the finite sections discretization and results on spectral and pseudospectral approximation.

We completely characterize the ranks of A - B and $A^{1/2}-B^{1/2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m=rank(A^{1/2}-B^{1/2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of $rank(A^{1/2}-B^{1/2})$. For...

We give some fixed point theorems for firmly pseudo-contractive mappings defined on nonconvex subsets of a Banach space. We also prove some fixed point results for firmly pseudo-contractive mappings with unbounded nonconvex domain in a reflexive Banach space.

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is defined to be spectrally bounded if there is a constant M ≥ 0 such that r(Tx) ≤ Mr(x) for all x ∈ E, where r(·) denotes the spectral radius. We study some basic properties of this class of operators, which are sometimes analogous to, sometimes very different from, those of bounded operators between Banach spaces.