My research area lies in the intersection of Operator Theory and Ergodic Theory/Measurable Dynamics. In particular, I study Noncommutative Ergodic Theory, which looks at the convergence of ergodic averages of operators on semifinite von Neumann algebras. There are numerous analogies between measure theory and von Neumann algebras that make work here actually possible. My primary research interests in this area are in the study of the individual (pointwise) convergence of subsequential and weighted (and subsequential weighted) averages of a positive Dunford-Schwartz operator on a von Neumann algebra (and its associated noncommutative L_p-spaces). I also have interest in the noncommutative analogues of harmonic analysis, probability theory, and functional analysis/operator space theory, which are all also active areas of mathematics. Recently, I have also been looking into the methods of factorization of operators between L_p-spaces (like Rosenthal's and Nikishin's Theorem), but still have yet to do much here.
Suppose we ran an experiment and took measurements at consistent time intervals, and we were interested in the average value these measurements take over a large period of time. Consideration of the subsequential averages would become important if errors occurred that forced some amount measurements to need to be omitted - to take an analogy from my advisor, maybe the person taking the measurements was extremely tired causing them to nod off at unexpected times and forget to write some things down. By just looking at the data, we may not know there are errors just by looking at the data, so we would then have to take the averages of what was written down. This gives rise to numerous questions: would the subsequential averages act like the same averages over long periods of time? Given the sequence of running averages, if the averages including all terms would have converged, would the subsequential averages converge too? If so, do they converge to the same thing?
Weighted ergodic theorems can arise in many different ways. In the analogy above, for example, the tired and overworked worker could have just made numerous errors and may have written down numerous wrong values - maybe at the 10 second mark they wrote down 7 but the actual measurement was 11 (but someone who wasn't there would never even know a mistake was made). Maybe it wasn't even the fault of the worker, but of the equipment! In reality one can't take measurements with 100% accuracy, instead having to make approximations - for example, a common bathroom scale often only your weight to within around 0.1 pounds. Hence, one may already have errors when computing the running averages, which can be accounted for by using weights. Other problems may just naturally be transformed into a problem that may be solved using weighted averages. For example, weighted averages can also arise when studying subsequential averages - it can be shown that, along certain subsequences (namely, ones with positive density), convergence of the subsequential averages is equivalent to the convergence of the averages when weighted by 1's and 0's (with a weight of 1 if that power of the operator is in the subsequence, and a 0 if that power is not).
In classical ergodic theory, one studies the averages of f(T^n(x)) as n tends to infinity, where X is a probability space, T is a measure-preserving map taking X to itself, x is in X, and f is a measurable real or complex valued function on X (usually living in some Lp-space). One modification that seems simple to make would be to consider functions f whose outputs are vectors (or, more generally, matrices) instead of numbers, as such outputs could represent multiple measurements being taken at a time, a position in 3-dimensional space, or a probability matrix, for example. It turns out that, although a seemingly simple change, this can be a lot harder to work with than one would expect. However, instead of replacing the outputs of the functions with matrices, one could instead replace the entire function itself with matrices (or, more generally, with a class of operators on a Hilbert space). As such, one would need to replace the movement of the input space (i.e. taking f(x) to f(T(x)) for x in X) with an operator on the space of operators (matrices, operators, etc.) under consideration.
Classical (i.e. commutative) ergodic theory occurs when having L_∞(X) (the bounded measurable functions) act on L_2(X), where for a fixed bounded measurable function g, one takes an L_2(X)-function f to the function M(g)f given by M(g)f(x)=g(x)f(g), for x in X. This assignment M(g) is linear and continuous as an operator on L_2(X), allowing one to view L_∞(X) as a subspace of the bounded operators on L_2(X). In this viewpoint, L∞(X) is actually a von Neumann algebra on L_2(X). Note that, when the outputs of f are n-dimensional vectors instead of just scalars, the expression M(g)f(x)=g(x)f(x) still makes sense when g(x) is an n-by-n matrix for every x in X. If g is a bounded scalar-valued function, it still acts on the n-dimensional vector-output L_2(X) as well by multiplying each component of the vector f(x) by g(x) (this corresponds to using M(g)=g(x)I_n, where I_n denotes the n-by-n identity matrix).
Below is a list of my publications and preprints:
M. O'Brien, "Noncommutative ergodic theorems for T-admissible processes", under preparation (2024)
M. O'Brien, "A noncommutative weak type maximal inequality for modulated ergodic averages with general weights", To appear in Colloquium Mathematicum (2024); https://arxiv.org/abs/2302.04466
M. O'Brien, "On subsequential averages of sequences in Banach spaces", Real Analysis Exchange (2023); https://doi.org/10.14321/realanalexch.48.2.1665637941
M. O'Brien, "Noncommutative Wiener-Wintner type ergodic theorems", https://arxiv.org/abs/2106.08906, Studia Mathematica (2023); DOI: 10.4064/sm211209-26-8
M. O'Brien, "Some noncommutative subsequential weighted individual ergodic theorems", https://arxiv.org/abs/2103.16784, Infinite Dimensional Analysis, Quantum Probability (2021); DOI: 10.1142/S0219025721500181