Computable copies of ℓp

Suppose p is a computable real so that p ≥ 1. It is shown that the halting set can compute a surjective linear isometry between any two computable copies of Rᵖ. It is also shown that this result is optimal in that when p /= 2 there are two computable copies of Rᵖ with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, Rᵖ is ∆⁰-categorical and is computably categorical if and only if p = 2. It is also demonstrated that there is a computably categorical Banach space that is not a Hilbert space. These results hold in both the real and complex case

The isometry degree of a computable copy of ℓp\ell^p

When $p$ is a computable real so that $p \geq 1$, the isometry degree of a computable copy $\mathcal{B}$ of $\ell^p$ is defined to be the least powerful Turing degree that computes a linear isometry of $\ell^p$ onto $\mathcal{B}$. We show that this degree always exists and that when $p \neq 2$ these degrees are precisely the c.e. degrees

Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p, where the constant C depends only on p and the modulus of uniform convexity. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. We clarify the relationship between bounds on the number of epsilon-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp