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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)?

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You can call it a diagonalizable Schur matrix. – percusse Feb 9 '15 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.

As for a name, in my opinion "subunitary" would seem appropriate. But that is just a guess and not based on knowledge.

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