Question

In an expository article clarifying this theorem to non-specialists, Paul Halmos asks “What does [this theorem] say?” An especially canonical form of this theorem yields an equivalence to a multiplication operator on a space given by a direct integral. The unitary case of this theorem may be quickly generalized by applying the Cayley transform and invoking a “multiplication theorem.” In functional analysis, this result is usually proven in a sequence of generalizations starting from compact operators and moving towards the unbounded (*) Hermitian case. (-5[1])When stated (10[1])for self-adjoint (-5[1])operators, (10[1])this result decomposes an operator (10[1])as an integral (10[1])with respect to a (10[1])projection-valued measure. By this theorem, an operator T induces a resolution of the identity on a set denoted “sigma (10[1])of T.” For 10 points, identify this theorem in linear algebra named for the set of a (-5[1])matrix’s eigenvalues. ■END■ (10[8])

Buzzes

Summary