Weighted Shifts and Banach Algebras of Power Series

Grabiner, Sandy

doi:10.2307/2373659

Cited by 35 publications

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“…In fact, in [2], the relationship between Sheffer operators, differentiation-invariant operators, and shift-invariant operators, as well as the importance of the spectrum in the characterization of isomorphisms, was shown. Similar questions were studied in the papers [6,7,8]; for differentiation-invariant operators, see [1,2,12,16].In the present paper, we consider shift-invariant operators on infinite power series spaces. Necessary and sufficient conditions for an operator to be continuous are given for any infinite power series space.…”

mentioning

confidence: 63%

Operators commuting with the shift on sequence spaces

Prada

2004

International Journal of Mathematics and Mathematical Sciences

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A complete characterization of shift-invariant operators that are isomorphisms is given in certain sequence spaces. Also given is a sufficient condition for an operator commuting with a shift-invariant operator to be shift invariant.2000 Mathematics Subject Classification: 47B37, 46A45. Introduction.There is a long-standing interest in linear continuous operators commuting with the right shift operator, weighted or not. The study of these operators is closely related to the study of operators commuting with the differentiation operator (weighted left shift). Several authors have treated topics connected with these operators; for instance, for unweighted shifts, a good reference is [10], while a good source for weighted shift operators is the papers [13,14,15] and the book of Halmos [9].The concrete problem of determining the spectrum of a weighted right shift operator was studied mainly by Gellar [3,4,5]. There is a strong relationship between the spectrum of such an operator and the question of whether or not an operator which commutes with it is an isomorphism. In [17] the spectrum of the differentiation operator on certain sequence spaces was computed directly, although it could have been deduced using [4, theorem 10].In the so-called umbral calculus appears the concept of a delta operator [2], which is invariant by differentiation and so connected with shift-invariant operators. In fact, in [2], the relationship between Sheffer operators, differentiation-invariant operators, and shift-invariant operators, as well as the importance of the spectrum in the characterization of isomorphisms, was shown. Similar questions were studied in the papers [6,7,8]; for differentiation-invariant operators, see [1,2,12,16].In the present paper, we consider shift-invariant operators on infinite power series spaces. Necessary and sufficient conditions for an operator to be continuous are given for any infinite power series space. Also given is a complete characterization of isomorphisms when the space is nuclear and a projective limit of Banach algebras. In addition, we give a sufficient condition for an operator commuting with a shift-invariant operator also to be shift invariant.

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confidence: 63%

Operators commuting with the shift on sequence spaces

Prada

2004

International Journal of Mathematics and Mathematical Sciences

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“…If w is a positive continuos decreasing weight in R + such that {w(n)} n≥1 is logarithmically concave, then there exists a log-concave function W in R + satisfying equation (8).…”

Section: 1mentioning

confidence: 99%

An extension of a theorem of Domar on invariant subspaces

Gallardo-Gutiérrez¹,

Partington²,

Rodriguez³

2017

Acta Sci. Math.

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Abstract. A remarkable theorem of Domar asserts that the lattice of the invariant subspaces of the right shift semigroup {Sτ } τ ≥0 in L 2 (R + , w(t)dt) consists of just the "standard invariant subspaces" whenever w is a positive continuous function in R + such that(1) log w is concave in [c, ∞) for some c ≥ 0, (2) lim t→∞ − log w(t) t = ∞, and limWe prove an extension of Domar's Theorem to a strictly wider class of weights w, answering a question posed by Domar in [6].

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“…It is not immediately obvious that there can be non-standard ideals in R(w), and, indeed, much of the literature about these algebras shows that if the weight function w is``nice'' (log-concave (see [7]), star-shaped (see [5], [14], and [15]), etc.) then there are only standard ideals.…”

Section: Semi-direct Decompositionsmentioning

confidence: 99%

Prime-like Elements and Semi-direct Products in Commutative Banach Algebras

Thomas

1997

Journal of Functional Analysis

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We develop results which show that elements in the radical of a commutative Banach algebra are often precluded from having prime-like properties if we avoid certain exceptional situations involving torsion elements. This makes the proof of the Singer-Wermer conjecture conceptually much clearer. It also motivates the definition of an element having regular powers and allows us to strengthen our previous results concerning necessary conditions for a commutative Banach algebra A to be the semidirect product of some subalgebra together with a specified principal ideal sA > , or, equivalently, concerning necessary conditions for there to be an algebraic splitting of the short exact sequencefor some given element s in A. In particular, we show that if A is a radical Banach algebra and s has regular powers then no such splitting is possible.1997 Academic Press BACKGROUND AND NOTATIONThis paper continues an investigation of the structure of the radical of a commutative Banach algebra which was pioneered by G. R. Allan [1] and J. Esterle [6], and which the author continued in [11] and [12]. Our attempt here is both to unify some of the existing results by focusing our attention on non-nilpotent elements with prime-like properties in the radical, and to improve existing theorems, such as those in [12]. Our main theorem is Theorem 4.1 in Section 4; this section also contains some interesting examples.We need to make some definitions and explain our notation. Let A denote a commutative algebra over the complex field. For the following definitions no topology is needed, but we will eventually specialize to the article no. FU963020 44

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Weighted Shifts and Banach Algebras of Power Series

Cited by 35 publications

References 20 publications

Operators commuting with the shift on sequence spaces

Operators commuting with the shift on sequence spaces

An extension of a theorem of Domar on invariant subspaces

Prime-like Elements and Semi-direct Products in Commutative Banach Algebras

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