Description:

Title: Lexicographic functional calculus

Abstract:  If A is a self-adjoint operator on a complex Hilbert space and f is a continuous function of a real variable, one can form the (normal) operator f(A) via the functional calculus construction, which appears in many areas of mathematics and physics.  Though the continuity of f implies a kind of continuity of the "operator function'' $A \mapsto f(A)$, it is generally not true that $A \mapsto f(A)$ is $C^k$ when f is $C^k$ ($k \geq 1$).  The (lack of) regularity of operator functions has been investigated since at least the 1950s;  the main tool used in such studies is called a ``multiple operator integral'' (MOI).  In my talk, I shall discuss MOIs, their relationship to questions about the regularity of operator functions, and a new framework for MOI-type objects called lexicographic functional calculus.