Commuting spectral measures on Hilbert space

Wermer, John

doi:10.2140/pjm.1954.4.355

Cited by 96 publications

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“…Since C*(T) is isometrically isomorphic to C(a(T)), it follows from part (2) of Theorem 3.4 that the mapping <p-> lim V"~x<p(T)V" is a homeomorphic isomorphism from C(o(T)) onto appr(5)". It follows from [47] that S is similar to a normal operator, which, being approximately similar to T, must be approximately equivalent to T. The converse is trivial.…”

Section: Corollary 23 If S Isa Separable Subset Ofb(h) Then #"( §)mentioning

confidence: 97%

An asymptotic double commutant theorem for 𝐶*-algebras

Hadwin¹

1978

Trans. Amer. Math. Soc.

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Abstract. An asymptotic version of von Neumann's double commutant theorem is proved in which C*-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and «■ is a representation of the C*-algebra generated by I, then ir(T) is similar to a normal (subnormal) operator.1. Introduction. One of the reasons for the success of the theory of von Neumann algebras is J. von Neumann's double commutant theorem [46], which gives an alternate description of the weak closure of a selfadjoint algebra of operators. It is the purpose of this paper to prove an asymptotic version of the double commutant theorem that gives an alternate description of the norm closure of a selfadjoint algebra of operators. This asymptotic double commutant theorem helps to unify asymptotic versions of various operator-theoretic concepts (e.g., similarity, reflexivity, reductivity). Applications are made to the study of operators that are similar to normal (or subnormal) operators. Also a proof is given that a strongly reductive, nonseparable, commutative, norm closed algebra of operators is selfadjoint.Throughout, H denotes a separable, infinite-dimensional complex Hubert space, B(H) denotes the set of operators (bounded linear transformations) on H, and %(H) denotes the set of compact operators on H. Also § denotes a separable, nonempty subset of B(H). However, in §8 the separability

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Section: Corollary 23 If S Isa Separable Subset Ofb(h) Then #"( §)mentioning

confidence: 97%

An asymptotic double commutant theorem for 𝐶*-algebras

Hadwin¹

1978

Trans. Amer. Math. Soc.

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“…is bounded, it is easy to see that the problem has an affirmative solution. For then, by a theorem due to Lorch and Mackey which is proved in [4], there exists an invertible A in B such that…”

Section: And C(z)ementioning

confidence: 99%

A characterisation and two examples of Riesz operators

Gillespie

West

1968

Glasgow Math. J.

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Introduction.A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. § 3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.Our nomenclature will be that of [5] and [6]. Let us recall that X\s a Banach space and that 8 , <£,

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“…The commuting spectral measures on Hilbert spaces were first studied by Wermer [13]. We shall quote this remarkable result.…”

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confidence: 90%

“…Preliminaries. For convenience we shall summarize here some definitions and results concerning the Bade theory of multiplicity for Boolean algebras of projections on Banach spaces (see [2] and [3] and the general theory of spectral operators (see [4], [6], [7], [8], and [13])).…”

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Operators Commuting with Boolean Algebras of Projections of Infinite Multiplicity

Tzafriri¹

1967

Transactions of the American Mathematical Society

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The algebra of operators commuting with a Boolean algebra of projections of finite uniform multiplicity (in the sense of Bade [3]) has been studied by Foguel [10] and the author [12]. The aim of the present paper is to show that most properties described in these two papers can be extended (sometimes, under additional conditions) to the algebra S of all the operators which commute with a complete countably decomposable Boolean algebra of projections containing no projections of infinite uniform multiplicity. It should be mentioned that one cannot get interesting general results in the case in which there are projections of infinite uniform multiplicity in the Boolean algebra of projections, since every operator on a Banach space commutes with the Boolean algebra of projections composed from the identities 0 and 7.We shall start by proving that for every operator AeS. there is a sequence of projections belonging to the above mentioned Boolean algebra of projections which increases to the identity and A multiplied by any element of this sequence is a spectral operator. Relying on this result we study the spectrum of operators of 5 and give a necessary and sufficient condition for such an operator to be spectral.In the following section we generalize Theorem 8 of [12] showing that in H a strong limit of spectral operators on a Hilbert space is spectral provided that they are of the same finite type and their resolutions of the identity are uniformly bounded. Adequate examples elucidate why we require the boundedness of the type of the spectral operators in most theorems.

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Commuting spectral measures on Hilbert space

Cited by 96 publications

References 1 publication

An asymptotic double commutant theorem for 𝐶*-algebras

An asymptotic double commutant theorem for 𝐶*-algebras

A characterisation and two examples of Riesz operators

Operators Commuting with Boolean Algebras of Projections of Infinite Multiplicity

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