Dynamics of Linear Operators

By definition, a continuous linear operator, T, on a topological k-vector space, X (with k = R or C), is “hypercyclic” if there is a vector, x, in X, such that the set of all vectors Tⁿ(x), with n running through the natural numbers, is dense in X. And T is “supercyclic” if there’s an x with the property that the set of all the cTⁿ(x), with c running through k and n in N, is dense in X. Finally, and familiarly, x is cyclic if k[T]x is dense in X. These are the players at the top on the list of the cast of characters appearing in Dynamics of Linear Operators, by Frédéric Bayart and Étienne Matherton. They’re introduced on page 1, and then the first proposition in the book appears, stating that if X ≠ {0} and dim(X) < ∞, there are no hypercyclic operators on X.

In most cases of interest X will be a Fréchet space: X is locally convex, complete, and metrizable. Thus it’s clear from the outset (we’re only on p. 2, after all) that functional analysis is on the scene in a big way: Banach spaces, even Hilbert spaces are waiting in the wings.

Bayart and Matherton race on to prove (pp. 2–3) the transitivity theorem of (George David) Birkhoff, to the effect that if X is a separable, complete, and metrizable topological vector space then T is hypercyclic if and only if T is topologically transitive: for any non-empty open sets, U, V, in X, there is an iterate Tⁿ such that Tⁿ(U) intersects V; and then the set of all hypercyclic vectors for T is a dense G_d subset of X.

Well, this is manifestly very serious stuff for the unsuspecting: we’re only on p. 3 and are already climbing to 30,000 feet — fast. At the same time, it’s clear as wodka that the material Bayart and Matherton present to us ought to be accessible to, say, a decently prepared young graduate student willing to work hard (is the second stipulation redundant?). The authors say explicitly, in fact, that “[they] hope that various kinds of readers will find [the] book useful, e.g. PhD students, specialists in the area and non-specialists wanting to get a flavour of the subject … it should be accessible to a rather large audience, including graduate students with an interest in functional analysis.” That says it all, really: specialists should be keen on this stuff because of the relative youth of the subject of dynamics of linear operators and the evidently high current level of activity in this area. Outsiders should be attracted to this material simply because of the obvious promise the study of dynamics of linear operators holds vis à vis applications across the mathematical spectrum (if I may be excused an easy and egregious pun) — additionally the intrinsic elegance of all things functional analytic should draw in a lot of dilettantes.

And here is what one can expect, in the way of a sampling of topics in the pages of the book: “Hypercyclicity everywhere” (including, “2.4 There are many hypercyclic operators,” “2.5 There are few hypercyclic operators” [only in functional analysis, eh?], and then, yes, “Linear dynamics is complicated” [No kidding ... ]), “Weakly mixing operators,” “Ergodic theory and linear dynamics’” [not surprisingly], and then on to “Beyond hypercyclicity,” and then… Well, see for yourself.

From my own parochial point of view it’s very exciting that Gaussian measures are discussed in the context of ergodic theory and that the penultimate chapter in the book is devoted to the Riemann zeta function. Given that qua functional analysis I belong to the dilettante class, this material by itself qualifies as a sufficient reason to delve deeper and deeper: might exciting arithmetical possibilities be hiding in the shadows?

To be sure, Dynamics of Linear Operators is very compactly written, but it’s quite accessible: the dense appearance of the presentation is largely due to the (welcome) wealth of detail in the proofs — Bayart and Matheton are thorough to a fault (if there is such a thing), but the pacing is good, if a little fast. The book also comes equipped with sections titled “Comments and exercises,” appearing at the end of each of the twelve chapters. These sections are gems in their own right: the comments are wonderful and often revealing and once more evince the authors’ high level of scholarship; the exercises are, well, fruity. There are four appendices: complex analysis, function spaces, Banach space theory, and spectral theory.

It’s an impressive work, filled with serious and fascinating mathematics. Each of the three audiences defined by Bayart and Matherton should thrive on it.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.