Originally posted on What's new:

One of the basic problems in the field of operator algebras is to develop a functional calculus for either a single operator $latex {A}&fg=000000$, or a collection $latex {A_1, A_2, \ldots, A_k}&fg=000000$ of operators. These operators could in principle act on any function space, but typically one either considers complex matrices (which act on a complex finite dimensional space), or operators (either bounded or unbounded) on a complex Hilbert space. (One can of course also obtain analogous results for real operators, but we will work throughout with complex operators in this post.)

Roughly speaking, a functional calculus is a way to assign an operator $latex {F(A)}&fg=000000$ or $latex {F(A_1,\ldots,A_k)}&fg=000000$ to any function $latex {F}&fg=000000$ in a suitable function space, which is linear over the complex numbers, preserve the scalars (i.e. $latex {c(A) = c}&fg=000000$ when $latex {c \in {\bf C}}&fg=000000$), and should be either an exact or approximate homomorphism in the sense that

$latex \displaystyle FG(A_1,\ldots,A_k) = F(A_1,\ldots,A_k) G(A_1,\ldots,A_k), \ \ \ \ \ (1)&fg=000000$

should hold either exactly or approximately. In the case when the $latex {A_i}&fg=000000$ are self-adjoint operators acting on a Hilbert space (or Hermitian matrices), one often also desires the identity

$latex \displaystyle \overline{F}(A_1,\ldots,A_k) = F(A_1,\ldots,A_k)^* \ \ \ \ \ (2)&fg=000000$

to also hold either exactly or approximately. (Note that one cannot reasonably expect (1) and (2) to hold exactly for all $latex {F,G}&fg=000000$ if the $latex {A_1,\ldots,A_k}&fg=000000$ and their adjoints $latex {A_1^*,\ldots,A_k^*}&fg=000000$ do not commute with each other, so in those cases one has to be willing to allow some error terms in the above wish list of properties of the calculus.) Ideally, one should also be able to relate the operator norm of $latex {f(A)}&fg=000000$ or $latex {f(A_1,\ldots,A_k)}&fg=000000$ with something like the uniform norm on $latex {f}&fg=000000$. In principle, the existence of a good functional calculus allows one to manipulate operators as if they were scalars (or at least approximately as if they were scalars), which is very helpful for a number of applications, such as partial differential equations, spectral theory, noncommutative probability, and semiclassical mechanics. A functional calculus for multiple operators $latex {A_1,\ldots,A_k}&fg=000000$ can be particularly valuable as it allows one to treat $latex {A_1,\ldots,A_k}&fg=000000$ as being exact or approximate scalars simultaneously. For instance, if one is trying to solve a linear differential equation that can (formally at least) be expressed in the form

$latex \displaystyle F(X,D) u = f&fg=000000$

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