Page 348 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 31: Finite Dimensional Spectral Theory

Notes
and x x A … (2)
n k
we can find a (y ) of (y ) s.t.
n
n k
y x = y x x x
n k n k n k n k

y x x x
n k n k n k
0 as n .

A is compact.
This completes the proof of the theorem.

31.2 Summary

 If T is an arbitrary operator on a finite dimensional Hilbert space H, then the spectrum of
T namely (T) is a finite subset of the complex plane and the number of points in (T) does
not exceed the dimension n of H.

 Let T be bounded linear operator on a Hilbert space H. Then a scalar is called an eigenvalue
of T if there exists a non-zero vector x in H such that Tx = x.
 The closed subspace M is called the eigenspace of T corresponding to the eigenvalue .

 The set of all eigenvalues of an operator is called the spectrum of T. It is denoted by (T).
 The spectral resolution of the normal operator on a finite dimensional non-zero Hilbert
space is unique.

 A subset A in a normed linear space N is said to be relatively compact if its closure A is
compact.

31.3 Keywords

Eigenspace: The closed subspace M is called the eigenspace of T corresponding to the eigenvalue .
Eigenvalues: Let T be bounded linear operator on a Hilbert space H. Then a scalar is called an
eigenvalue of T if there exists a non-zero vector x in H such that Tx = x.
Eigenvalues are sometimes referred as characteristic values or proper values or spectral values.
Eigenvectors: If is an eigenvalue of T, then any non-zero vector x H such Tx = x, is called a
eigenvector.
Similar Matrices: Let A, B are square matrix of order n over the field of complex number. Then
B is said to be similar to A if there exists a n × n non-singular matrix C over the field of complex
numbers such that
B = C AC.
–1
Spectrum of an Operator: The set of all eigenvalues of an operator is called the spectrum of T and
is denoted by (T).
Total Matrices Algebra: The set of all n × n matrices denoted by A is complex algebra with
n
respect to addition, scalar multiplication and multiplication defined for matrices.
This algebra is called the total matrices algebra of order n.

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