A Course in Functional Analysis

Conway, John B.

doi:10.1007/978-1-4757-3828-5

Cited by 724 publications

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“…is a self-adjoint operator we call {EA(t)\t G R} the spectral nest of A (see [1] for details). There is no loss of generality in assuming that every nest JA is the spectral nest of a positive invertible operator.…”

mentioning

confidence: 99%

Descriptions of nest algebras

Anoussis¹,

Katsoulis²

1990

Proc. Amer. Math. Soc.

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Abstract.Every operator in the algebra of a continuous nest A~ can be factored as a product of two operators which belong to certain diagonal disjoint ideals of AlgyT. This factorization leads to a new description of nest algebras.The main result of this paper is that every operator in the algebra of a continuous nest can be factored as a product of two operators which belong to certain diagonal disjoint ideals of that algebra. As a consequence, a new description of nest algebras is given, in terms of factorization criteria.Let yf be a nest (i.e., a totally ordered complete set of projections containing 0 and / ) on a separable Hilbert space %A. We denote by Alg JA the algebra of all operators in B(%A) that leave invariant every element of JA. Also, by JA' we will denote the diagonal of Alg JA, i.e., the set of all operators in Alg JA such that their adjoints also belong to Alg JA. Finally, we call JA" to be the core of Alg JA .Ideals of nest algebras and in particular ideals which are diagonal disjoint have been studied by a number of authors [4,5,9]. For example, the Jacobson Radical £%A-of Alg JA is a diagonal disjoint two-sided ideal of Alg JA and one of the main results of [9] is a characterization of this ideal. Let us give some definitions.We shall call a (not necessarily finite) set {Pn\a G sé , sé G N} of intervals P = M -N , M , N G JA, of the nest JA, a partition of JA if the intervals are pairwise orthogonal and the sum J2aÇjf Pn = / in the strong operator topology. Given a nest JA , Larson s ideal AH™ is the collection of all operators X in Alg JA for which, given e > 0, there exists a partition {PJa G sé} of JA such that ||P XP || < e for all a G sé .If we restrict all partitions to be finite sets of intervals we will obtain the Jacobson Radical AP r of Alg JA [9]. Since, however, infinite partitions can arise in general, AAlf-will usually contain ¿A? r . The ideal AA?0^. plays an important role in the theory of nest algebras and their similarities [6]. Recently, J. Orr [11] proved that, for an arbitrary nest JA, AAl™ is the greatest diagonal disjoint ideal of Algyf . One of the key lemmas in the proof of the above result

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confidence: 99%

Descriptions of nest algebras

Anoussis¹,

Katsoulis²

1990

Proc. Amer. Math. Soc.

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“…However, this requires a recasting of our framework in a functional analytical setting, using results and concepts from the theory of Hilbert spaces and C*-algebras (see e.g. [3,16]), which would deserve a separate treatment.…”

Section: Discussionmentioning

confidence: 99%

Probabilistic -calculus and Quantitative Program Analysis

Pierro

Hankin

Wiklicky

2005

Journal of Logic and Computation

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We show how the framework of probabilistic abstract interpretation can be applied to statically analyse a probabilistic version of the λ-calculus. The resulting analysis allows for a more speculative use of its outcomes based on the consideration of statistically defined quantities. After introducing a linear operator based semantics for our probabilistic λ-calculus Λp, and reviewing the framework of abstract interpretation and strictness analysis, we demonstrate our technique by constructing a probabilistic (first-order) strictness analysis for Λp.

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“…Before we present the proof of Theorem 1.3, we recall some facts from the spectral theory of bounded linear operators (see Chapter VII of [3]) and we establish some lemmas.…”

Section: Proofmentioning

confidence: 99%

“…where ins Γ and out Γ denote the inside of Γ and the outside of Γ, respectively (see Chapter VII, Section 6.9 of [3]). The bounded linear operator P σ has properties…”

Section: Proofmentioning

confidence: 99%

“…Then every λ ∈ σ(T )\{0} is an eigenvalue of T and for every > 0 the set {λ ∈ σ(T ) | |λ| ≥ } ⊂ σ p (T ) is finite. It is known that each λ ∈ σ(T ) \ {0} is a pole of the resolvent (z − T ) −1 and the generalized eigenspace P {λ} (X) is finite dimensional (see Corollary 7.8 of [3]). Now suppose that σ ⊂ σ(T )\{0} is a finite set.…”

Section: Proofmentioning

confidence: 99%

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