On absolutely continuous functions and the well-bounded operator

Sills, William

doi:10.2140/pjm.1966.17.349

Cited by 5 publications

(9 citation statements)

References 13 publications

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“…Given an operator T on a dual Banach space X with a C(σ ) functional calculus, we directly extend this to B(σ ) and then use the extended functional calculus to construct a spectral measure which will provide an integral representation for the operator. The extension method is due to Sills, who proved in [11,Theorem 6.1] that a continuous homomorphism from a commutative Banach algebra U to L(X ) may be extended to a homomorphism from U * * to L(X ) when X is reflexive. He further stated the following theorem [11,Theorem 6.2], which is proved by essentially the same technique; we provide details of the proof here as we shall use the construction in what follows.…”

Section: C(σ ) Functional Calculusmentioning

confidence: 99%

“…In [4] the authors note that 'the situation when X = ∞ seems to be open'. We shall answer this question in Section 2, by using a technique due to Sills [11] to show that if X is a dual Banach space, then a C(σ ) functional calculus for an operator T ∈ L(X ) may always be extended to a B(σ ) functional calculus for T . As is usual in the study of such operators, we frame this result in the context of the rich theory of spectral, prespectral and finitely spectral operators.…”

Section: Introductionmentioning

confidence: 99%

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Functional Calculus Extensions on Dual Spaces

Terauds¹

2009

Bull. Aust. Math. Soc.

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In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.2000 Mathematics subject classification: primary 47B40.

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Section: C(σ ) Functional Calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Functional Calculus Extensions on Dual Spaces

Terauds¹

2009

Bull. Aust. Math. Soc.

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“…Let BV(J) denote the set of functions of bounded variation over J. If we define IIf Ili = If (b)I + varj f then 11 " Ili is a norm on BV(J) which makes this space into a Banach algebra [61].…”

Section: Bv(j)mentioning

confidence: 99%

Spectral Theory of Linear Operators

Pugachev¹,

Sinitsyn²

1999

Lectures on Functional Analysis and Applications

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SummaryThis thesis is concerned with the relationship between spectral decomposition of operators, the functional calculi that operators admit, and Banach space structure.The deep connection between the first two of these concepts has long been known.The thesis is organised as follows. Chapter 1 is an introduction to the concepts, ideas and constructions that will be used through at this thesis. Particularly we consider numerical range and hermitian operators which have a critical role in all this thesis.In Chapter 2 we give a brief overview of some of the theory of (strongly) normal (equivalent) operators. Developing the properties of (strongly) normal (equivalent) operators we will show that the possession of a functional calculus on the spectrum of T is equivalent to T' being scalar type prespectral of class X, thus answering a and if either X does not contain a copy of co, or if U and V are decomposable in X, then the representation is unique. We also explore some properties of AC-operators by applying the theory of (Foia §) decomposable operators.Since 1954 the problem of giving sufficient conditions for the sum and product of two commuting spectral operators to be spectral has attracted attention. The boundedness of the Boolean algebra of projections generated by the two resolutions of the identity is critical. then the Boolean algebra of projections generated by £ and F is also bounded.As a consequence of this he showed that the sum and product of two commuting spectral operators is also spectral in each of the above cases. We will show that the weakly closed algebra generated by the real and imaginary parts of a finite family of commuting scalar-type spectral operators on a Banach lattice not containing co, and on a closed linear subspace of a p-concave Banach lattice, where p< oo, is a W*-algebra, and that every operator in this algebra is a scalar-type spectral operator.

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“…However, if T is the weak operator topology, then the Assets each contain a unique projection. LEMMA [10] The spectral theorem 343…”

Section: Jamentioning

confidence: 99%

“…Furthermore, the exact conditions for a family of projections to be a decomposition of the identity are somewhat cumbersome. An alternative approach to that of Smart and Ringrose was introduced by Sills [10] who used a technique that involved first extending the AC functional calculus to the second dual, a space which contains idempotent elements.…”

Section: Introductionmentioning

confidence: 99%

The spectral theorem for well-bounded operators

Doust

Bozhou

1993

J Aust Math Soc A

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Well-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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On absolutely continuous functions and the well-bounded operator

Cited by 5 publications

References 13 publications

Functional Calculus Extensions on Dual Spaces

Functional Calculus Extensions on Dual Spaces

Spectral Theory of Linear Operators

The spectral theorem for well-bounded operators

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