Page 319 - DMTH502_LINEAR_ALGEBRA

Unit 29: Spectral Theory and Properties of Normal Operators

Since Notes

I = E 1ji E 2 2 j E mj m
1 j 2 j jm
each vector in V may be written in the form

E
= E 1j 1 2 j 2 ...E mj m . … (3)
j 1 ..., j m
Suppose j , ..., j , are indices for which = E E , ... E 0. Let
1 m 1j1 2j2 mjm
= II E nj .
i n
n i
Then = E ; hence there is a scalar c such that
ij i i
T = c , 1 i m.
1 i
For each T in , there exist unique scalars b such that
i
m
i i
T = b T
i 1
Thus

T = b T
i i
i
= b c .
i i
i
The function T b c i i
i i , is evidently one of the roots, say r or , and lies in V(r ). Therefore,
i
each non-zero term in (3) belongs to one of the spaces V(r ), ..., V(r ). It follows that V is the
1 k
orthogonal direct sum of V(r ), ...,V(r ).
l k
Corollary: Under the assumptions of the theorem, let P be the orthogonal projection of V on
j
V(r ) (1 j k). Then P P = 0 when i j,
j i j
I = P + ... + P ,
1 k
and every T in  may be written in the form
P
T
T = r j ( ) . … (4)
j
i
Definition: The family of orthogonal projections {P , ..., P } is called the resolution of the identity
1 k
determined by , and (4) is the spectral resolution of T in terms of this family.
Although the projections P , ..., P , in the preceding corollary are canonically associated with the
1 k
family , they are generally not in  nor even linear combinations of operators in ; however,
we shall show that they may be obtained by forming certain products of polynomials in elements
of .
In the study of any family of linear operators on an inner product space, it is usually profitable
to consider the self-adjoint algebra generated by the family.
Definition: A self-adjoint algebra of operators on an inner product space V is a linear
sub-algebra of L(V, V) which contains the adjoint of each of its members.

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