Cayley transform

There is a beautiful dictionary between self-adjoint operators and unitary operators, and the Cayley transform is the translation key. It is the operator version of a familiar function of a complex number: the map that sends a real number to a point on the unit circle. For a self-adjoint operator A, the Cayley transform is U = (A - iI)(A + iI)^-1, and the result is always unitary.

Watch what it does to eigenvalues, which the spectral mapping theorem makes effortless. Each real eigenvalue lambda of A becomes the eigenvalue (lambda - i)/(lambda + i) of U. As lambda runs over the real line, that fraction runs over the unit circle (missing only the point 1, the image of lambda = infinity). So a self-adjoint operator, whose spectrum lives on the real axis, maps to a unitary operator, whose spectrum lives on the unit circle — a perfect spectral correspondence between the two natural homes of these operator classes.

The transform is reversible: A = i(I + U)(I - U)^-1 recovers the self-adjoint operator (as long as 1 is not an eigenvalue of U). This back-and-forth is genuinely useful. It lets you convert questions about self-adjoint operators into questions about unitary ones and vice versa, and it builds unitary operators with prescribed spectra. In infinite dimensions the same idea (with care about domains) is the standard device for studying unbounded self-adjoint operators through bounded unitary ones — the Cayley transform is a small finite-dimensional fact with a long reach.