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What the numerical range, Poncelet’s theorem, and Blaschke products have to do with it

In this talk, we look at three seemingly unrelated concepts: the numerical range of an operator, Poncelet’s theorem and Blaschke products. The connection appears through envelopes of lines. But Blaschke products can be thought of as hyperbolic polynomials, so perhaps there is a better way to think about the problem. We being by considering the numerical range of an n × n matrix A, defined by W (A) = {Ax, xi : x ∈ C ⁿ , |x| = 1}. With this in hand, we explain the connection and then explain a new way of looking at the problem. Generalizing this situation leads us to a class of well-known operators called compressions of the shift operator. Using this point of view, we obtain new insights into the numerical range of these operators.