Funktionalkalküle in mehreren Veränderlichen für stetige lineare Operatoren auf Banachräumen

Albrecht, Ernst

doi:10.1007/bf01637620

Cited by 21 publications

(22 citation statements)

References 11 publications

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“…The estimate exp(itΛ) = O(t s+r+2n ) as t → ∞ for s, r integers follows from a paper by Albrecht [1]. If s, r are half-integers then from [1] one can derive only exp(itΛ) = O(t s+r+3n ). Our estimate is a refinement of the last one.…”

Section: Elementary Operatorsmentioning

confidence: 99%

“…, r n , respectively. Also, let Λ : A → A be an elementary operator given by (1). If 0 ∈ {λ 1 µ 1 + · · · + λ n µ n | (λ 1 , .…”

Section: Elementary Operatorsmentioning

confidence: 99%

“…We give three independent applications. The first of them is an ascent estimate for an elementary operator (1), with generalized scalar a j and b j . For a linear mapping Λ : E → E on an arbitrary linear space E, the ascent asc(Λ) is defined as the least positive integer k such that ker(Λ k ) = ker(Λ k+1 ).…”

mentioning

confidence: 99%

“…If no such positive integer exists we set asc(Λ) = +∞. We estimate the ascent of the operator (1) in terms of the orders of a j , b j and the dimension of the set σ(a 1 , . .…”

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confidence: 99%

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Elementary operators on Banach algebras and Fourier transform

Arsenović

Kečkić

2006

Studia Math.

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Abstract. We consider elementary operators x → n j=1 a j xb j , acting on a unital Banach algebra, where a j and b j are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families {a j } and {b j }, i.e. a j = awhere all a 0. Introduction. The theory of generalized scalar operators on a Banach space was developed in [6]. Briefly, a ∈ A is a generalized scalar element of a unital Banach algebra A if it has real spectrum, and if for all real t, e ita ≤ C(1 + |t| s ), for some constant C depending only on a. Also, it is known that these two conditions are equivalent to the existence of a functional calculus for a, based on R. If s = 0, we say that such an element is pre-hermitian. In that case the condition of having real spectrum is not necessary. Also we can define pre-normal elements as elements of the form h+ik with h, k pre-hermitian. Many properties of pre-hermitian, pre-normal, and generalized scalar elements can be found in [6] and [5]. In Section 1 we review results concerning such elements, necessary for reading this note. In [13], a functional calculus for several commuting operators on a Banach space, using Fourier transform, was developed. In Section 2, we prove two results about L 1 behaviour of the Fourier transforms of a family of C ∞ cpt functions. These results have a central role in further applications to the theory of elementary operators on a unital Banach algebra.Section 3 contains applications of the results from Section 2 to elementary operators on a unital Banach algebra A, i.e. to mappings Λ : A → A of the form 2000 Mathematics Subject Classification: 47B48, 47B47, 42B10.

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Section: Elementary Operatorsmentioning

confidence: 99%

“…, r n , respectively. Also, let Λ : A → A be an elementary operator given by (1). If 0 ∈ {λ 1 µ 1 + · · · + λ n µ n | (λ 1 , .…”

Section: Elementary Operatorsmentioning

confidence: 99%

mentioning

confidence: 99%

“…If no such positive integer exists we set asc(Λ) = +∞. We estimate the ascent of the operator (1) in terms of the orders of a j , b j and the dimension of the set σ(a 1 , . .…”

mentioning

confidence: 99%

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Elementary operators on Banach algebras and Fourier transform

Arsenović

Kečkić

2006

Studia Math.

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“…However, the answer is positive for classes of systems satisfying the conditions of Theorem 4.5. Such examples are, for instance, the classes of systems having functional calculi [4], [1]. For the sake of simplicity we shall consider only a particular case (see [1] for details).…”

Section: Let T Be An M-decomposable Operator Then Its M-spectral Capmentioning

confidence: 99%