# What is the term for a matrix with spectral radius less than 1, with all eigenvalues of modulus = 1 associated with a 1x1 Jordan block?

I am looking at the matrices described in the title: matrices where all eigenvalues lie in the unit disc, and with the eigenvalues of modulus 1 having 1x1 Jordan blocks. My question is, is there a term for such matrices?

A related question is, since we lose the Jordan normal form in infinite dimensions, what is the appropriate infinite-dimensional analog of this condition (if there is one)?

-
You can call it a diagonalizable Schur matrix. –  percusse Feb 9 at 9:25

To the second question: You can check that your condition is equivalent to $$\limsup_{n \to \infty} \|A^n \| \leq 1.$$ If there is an eigenvalue > 1, it is clear that the above fails. If an eigenvalue = 1 has non-trivial Jordan block, you have $\|A^n\| \gtrsim n$, so the condition is violated.