Definable Operators on Hilbert Spaces

Abstract

Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize the definable linear operators on H as exactly the "scalar plus compact" operators.

An approximate Herbrand’s theorem and definable functions in metric structures

“…Then T is primitive as the Hilbert space axioms are universal and the axioms for infinite-dimensionality are existential. We must remark that we cannot work in the many-sorted setting for Hilbert spaces (as in [10]) because the axioms for the inclusion mappings are ∀∃; indeed, for n ≤ m, one must declare that the inclusion mapping I n,m : B n (H) → B m (H) is onto the set of elements of B m (H) of norm at most n.…”

“…, t n ( x)) is an atomic formula with I ϕ := I P . We let L ms denotes the many-sorted theory of Hilbert spaces used in [10]. Proof.…”

“…Using the main result of [10], we can give a different proof of this fact in the case that f is linear. Indeed, write f = λI + K, where K is a compact operator.…”

“…The main motivation for this paper comes from the study of definable functions in metric structures; this study was initiated by the author in [11], where a study of the definable functions in Urysohn's metric space was undertaken, and continued in [10], where the definable linear operators in (infinite-dimensional) Hilbert spaces were characterized. However, lacking any understanding of arbitrary definable functions in Hilbert spaces, we conjectured that they were, in some sense, "piecewise affine" in analogy with the classical case of an infinite vector space over a division ring.…”

Definable closure in randomizations

“…Continuous model theory in its current form is developed in the papers [BBHU] and [BU]. The papers [Go1], [Go2], [Go3] deal with definability questions in metric structures. Randomizations of models are treated in [AK], [Be], [BK], [EG], [GL], [Ke1], and [Ke2].…”