We define the eigenderivatives of a linear operator on any real or complex Banach space, and give a sufficient condition for their existence.
that the eigenvectors and eigenvalues of an operator "nearly equal" to K will typically be "nearly equal" to those of K . In this paper we explore this question and give a definition of the "eigenderivative," i.e. the "rates of change" of eigenvectors and eigenvalues.
Definition: Let X be a real or complex Banach space, and let End KX be a linear operator with a complete set of unit eigenvectors
where
identify the field over X . Suppose J is a linear operator such that for 1 The notation
End X denotes the linear operators mapping
converges absolutely, then the eigenderivatives exist.
Proof:
,, 11 11 sup
11
X be the completion of the vector space spanned by the simple functions
for positive integers n , with F . Let the be given by the 2 L norm, i.e.
- An unbounded operator, for which eigenderivatives exist.
Thus the requirements of the Definition are satisfied, and the eigenderivatives exist.
, depicted in the following:
- A bounded operator, for which eigenderivatives exist.
Thus the requirements of the Definition are satisfied, and the eigenderivatives exist.
They are
Proof: For ease of notation this proof will employ the abbreviations
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