Theorem (Analytic Fredholm Alternative) Let be a connected open subset of and be a separable Hilbert space. Suppose that is an analytic operator-valued function such that is compact . Then either
- does not exist .
- exists , where is discrete.
Theorem Let be a closed ideal of a C* algebra , then for any self-adjoint element , there exists an such that
Proof: Since is a closed ideal, is also a C* algebra and one can define a natural projection homomorphism , with . Fix an element , define a continuous function as follow:
Let , then is identical on . By continuous functional calculus, . And since can be approximated by polynomials, we know . The spectral radius , so we know is identical on the spectrum of , which implies . Therefore is in the kernel of , and hence in the ideal . Since and by the Gelfand Naimark isomorphism theorem, we know and hence . So is the we are looking for.
Note: this is a homework exercise given by Dr. Carlen.
Definition Let be a banach space, be the space of bounded linear operator from . If , then say a complex number is in the resolvent set of if has a bounded inverse. Then we define the resolvent of at to be . If , then is said to be in the spectrum of T.
We will learn some properties of the resolvent and the spectrum.