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Simultaneous stabilization in A ℝ ( )

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

We study the problem of simultaneous stabilization for the algebra A ℝ ( ) . Invertible pairs ( f j , g j ) , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that α f j + β g j is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since A ℝ ( ) has stable rank two, we are faced here with a different situation....

Solving power series equations. II. Change of ground field

Joseph Becker (1979)

Annales de l'institut Fourier

We study the effect of changing the residue field, on the topological properties of local algebra homomorphisms of analytic algebras (quotients of convergent power series rings). Although injectivity is not preserved, openness and closedness in the Krull topology, simple topology, and inductive topology is preserved.

Some results about Beurling algebras with applications to operator theory

Thomas Vils Pedersen (1995)

Studia Mathematica

We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying ∥ T n ∥ = O ( n β ) as n → ∞ for some β ≥ 0, then ∑ n = 1 ∞ ∥ ( 1 - T ) n x ∥ / ∥ ( 1 - T ) n - 1 x ∥ diverges for every x ∈ X such that ( 1 - T ) [ β ] + 1 x ≠ 0 .

Spaces of type H ∞ + C

Walter Rudin (1975)

Annales de l'institut Fourier

A simple theorem is proved which states a sufficient condition for the sum ot two closed subspaces of a Banach space to be closed. This leads to several analogues of Sarason’s theorem which states that H ∞ + C is a closed subalgebra of L ∞ . In these analogues, the unit circle is replaces by other groups, and the unit disc is replaced by polydiscs or by balls in spaces of several complex variables. Sums of closed ideals in Banach algebras are also studied.

Spectra of subnormal Hardy type operators

K. Rudol (1997)

Annales Polonici Mathematici

The essential spectrum of bundle shifts over Parreau-Widom domains is studied. Such shifts are models for subnormal operators of special (Hardy) type considered earlier in [AD], [R1] and [R2]. By relating a subnormal operator to the fiber of the maximal ideal space, an application to cluster values of bounded analytic functions is obtained.

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