### A Note on the Raical of a Banach Algebra.

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The Ritt and Kreiss resolvent conditions are related to the behaviour of the powers and their various means. In particular, it is shown that the Ritt condition implies the power boundedness. This improves the Nevanlinna characterization of the sublinear decay of the differences of the consecutive powers in the Esterle-Katznelson-Tzafriri theorem, and actually characterizes the analytic Ritt condition by two geometric properties of the powers.

We continue the paper [Ts] on the boundedness of polynomials in the Volterra operator. This provides new ways of constructing power-bounded operators. It seems interesting to point out that a similar procedure applies to the operators satisfying the Ritt resolvent condition: compare Theorem 5 and Theorem 9 below.

The intersection of the Gerschgorin regions over the unitary similarity orbit of a given matrix is studied. It reduces to the spectrum in some cases: for instance, if the matrix satisfies a quadratic equation, and also for matrices having "large" singular values or diagonal entries. This leads to a number of open questions.

Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a ${C}^{*}$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a ${C}^{*}$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the...

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