Page 341 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes that (T) . Further, this equation in has exactly n roots in complex field. If the equation has
repeated roots, then the number of distinct roots are less than n. So that T has an eigenvalue and
the number of distinct eigenvalue of T is less than or equal to n. Hence the number of elements
of (T) is less than or equal to n. This completes the proof of the theorem.

Example: For a two dimensional Hilbert space H, let B = {e , e } be a basis and T be an
1 2
operator on H given by the matrix

A = 11 12 … (1)
21 22
(i) If T is given by Te = e and Te = – e , find the spectrum T.
1 2 2 2
(ii) If T is an arbitrary operator on H with the same matrix representation, then
2
T ( + ) T + ( – ) I = 0
11 22 11 22 12 21
Sol:
(i) Using the matrix A of the operator T, we have

Te = e + e = e so that = 0 and = 1
1 11 1 21 2 2 11 21
Te = e + e = –e so that = –1 and = 0
2 12 1 22 2 1 12 22
0 1
Hence [T] = 11 12 .
21 22 1 0
For this matrix, the eigenvalue are given by the characteristic equation

1
1 = 0

2 + 1 = 0 = i so (T) = { i}.
(ii) Let us consider the eigenvalues of A, which are given by

11 12 = 0
21 22
2 – ( + ) + ( – ) = 0 … (2)
11 22 11 22 12 21
Since (2) is true for , we can take
T = I … (3)
From (2) and (3) we get
2
T – ( + ) T + ( – ) I = 0 … (4)
11 22 11 22 12 21
The operator T on H having as an eigenvalue satisfies equation (4).
Theorem 2: If T is an operator on a finite dimension Hilbert space, then the following statements
are true.
(a) T is singular 0 (T)

–1
(b) If T is non-singular, then (T) –1 (T )
–1
(c) If A is non-singular, then (ATA ) = (T)
(d) If (T) and if P is polynomial then P ( ) (P (T)).

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