Notes on Lewin Ch4: Spectral Theorem and Functional Calculus

Multiplication Operators

Let be Borel, and let be a locally finite Borel measure on . We set . Local finiteness implies that that .

• Example Let and let . Then , so every operator identifies under the obvious choice of basis with a matrix , with . Then is the element .

  • A diagonal matrix corresponds to the operator where .
  • Every Hermitian matrix can be identified with a diagonal matrix (possibly under a different basis).

• Let , then is the operator defined by

• Theorem Let .

  1. is closed.
  2. , where the essential range of is
  1. The eigenvalues of are the such that , with the corresponding eigenspace , the space of all square-integrable functions with support in the set , defined -a.e.
  2. is bounded iff .
  3. is self-adjoint iff is real-valued (bounded or not).

  • Bits Of The Proof. For , there is such that -a.e. Thus, , and the map is bounded (using ) and is an inverse for . For , there exists such that for all . Then with we have

    So cannot be invertible.

• Theorem 4.4: Spectral Theorem Let be self-adjoint on . Then there exists , a Borel set , a locally finite measure on , a real-valued locally bounded function , and an isomorphism such that

  One can take , and a finite measure on .

• Corollary .

• Corollary Every isolated point of the spectrum is an eigenvalue.

• Define for any such isomorphism. We need to show this is independent of the choice of .

• Theorem 4.8: Functional Calculus for bounded Borel functions Let be self-adjoint. There exists a unique map

  defined on the -algebra of bounded Borel functions on , with values in the algebra of bounded operators on , such that:

  1. it is a morphism of -algebras (-linear, preserves product and star operation).
  2. it is continuous, with .
  3. if with , then .
  4. if then .
  5. if and pointwise on , then for all .

• (Spectral measure) Let be a unit vector of . By the functional calculus, the map

  is a continuous linear form. If in addition we can write

  which shows that is a positive linear form on . Hence by Riesz–Markov, there is a unique Borel probability measure on such that

  With and from the Spectral Theorem, if is the corresponding unitary, we can write

  Therefore is the pushforward measure¹

  i.e. for every Borel set ,

  In words - is the cyllindrical projection on of the probability measure on . We then have iff has a moment of order two, and in this case . We also have .

• Corollary 4.10: Functional Calculus for locally bounded Borel functions Let be self-adjoint and let be a ocally bounded Borel function. Then defined above is independent of the isomorphism used to represent as a multiplication operator.

1. ¹The book writes but this is a little fast and loose with notation; the sum is a partial integration of that comes from the pushforward. The Proper Way is to define the slice measures , then we can write .