Power series of bounded linear operators

Question

If $f$ is a complex analytic function, one can define a matrix function $F$ using the Taylor series of $f$ by

$$ F(A) = f(0) + f'(0)\cdot x + f''(0)\cdot \frac{A^2}{2!} + \cdots $$

If the radius of convergence of $f$ is $r$, then the above matrix series will also converge for $\|A\|<r$, where $\|\cdot\|$ is a submultiplicative matrix norm.

Is this still valid if $A$ is a bounded linear operator instead of a matrix?

Answer 1

Yes it is still valid . An operator can be represented as a matrix . Assuming it's a normal matrix ,it must have an eigenbasis and if we transform it into that basis then it is a diagonal matrix.