Norm attaining multilinear forms on .
Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism : A → B such that (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications...
We show that a positive semigroup on with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the -scale, which may be of independent interest.
Let X be a closed subspace of , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like , 0 ≤ α < 1, in terms of , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie....
Harper’s operator is defined on by where . We show that the norm of is less than or equal to for . This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.
Let be a continuous function on and , , the convex set of selfadjoint operators with spectra in . If and , as an operator function, is Gateaux differentiable on while is Lebesgue integrable, then we have the inequalities where is the Gateaux derivative of .
The normal cohomology functor is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of -groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B) is normal projective and normal injective. It follows that there is a natural isomorphism between Hom of normal Shilov modules and that of their quotient modules, which is a new lifting theorem of normal Shilov...