Hey Gottfried,
did you ever thought about the spectrum of the Carleman matrix of \( \exp_b \)?
For finite matrices the spectrum is just the set of eigenvalues. However for an infinite matrix or more generally for a linear operator \( A \) on a Banach space the spectrum is defined as all values \( \lambda \) such that \( A-\lambda I \) is not invertible.
We saw that the eigenvalues of the truncated Carleman matrices of \( \exp \) somehow diverge. So it would be very interesting to know the spectrum of the infinite matrix. As this also has consequences on taking non-integer powers of those matrices.
Or is the spectrum just complete \( \mathbb{C} \) because \( A \) is not invertible itself?
did you ever thought about the spectrum of the Carleman matrix of \( \exp_b \)?
For finite matrices the spectrum is just the set of eigenvalues. However for an infinite matrix or more generally for a linear operator \( A \) on a Banach space the spectrum is defined as all values \( \lambda \) such that \( A-\lambda I \) is not invertible.
We saw that the eigenvalues of the truncated Carleman matrices of \( \exp \) somehow diverge. So it would be very interesting to know the spectrum of the infinite matrix. As this also has consequences on taking non-integer powers of those matrices.
Or is the spectrum just complete \( \mathbb{C} \) because \( A \) is not invertible itself?