Spectral measures and the Bade reflexivity theorem

Dodds, P. G.; Ricker, Werner J.

doi:10.1016/0022-1236(85)90032-1

Cited by 33 publications

(53 citation statements)

References 11 publications

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“…Here S8(C) is the cr-algebra of Borel subsets of C. The measure P T is called the resolution of the identity for T. So, L l (P T ) is always r(P r )-separable. Under certain completeness assumptions on the lc spaces L S (X) and L l (P T ) it turns out that L}{P T ) is isomorphic to the strong operator closed algebra in L S (X) generated by the range of the resolution of the identity for T; see [2]. Accordingly, this algebra of operators is necessarily separable for the strong and hence also the weak operator topology.…”

Section: Operator-valued Measuresmentioning

confidence: 99%

Separability of the L¹-space of a vector measure

Ricker

1992

Glasgow Math. J.

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Let 2 be a a-algebra of subsets of some set Q and let n :2-»[0, °°] be a a-additive measure. If 2(j*) denotes the set of all elements of 2 with finite jU-measure (where sets equal [i-a.e. are identified in the usual way), then a metric d can be defined in 2(ji) by the formula So, suppose that A' is a locally convex space (briefly, lcs), always assumed to be Hausdorff and sequentially complete. A a-additive map m:2-*X, where 2 is a a-algebra of subsets of some set Q, is called a (A*-valued) vector measure. A 2-measurable function / : Q -» C is called m-integrable if it is integrable with respect to the complex measure (m,x') :£>-» (m(E),x'), for £ e 2 , for every x' e X' (the continuous dual space of X), and if, for every £ e 2 , there exists an element of X, denoted by J" E fdm, which satisfies (J E fdm,x') = J E fd(m,x'), for every x' e A". The linear space of all m-integrable functions is denoted by L{m). Let 2. x denote the family of all continuous seminorms in X or, at least enough seminorms to determine the topology of X. Each q e 3. x induces a seminorm q{m) in L(m) via the formula https://www.cambridge.org/core/terms. https://doi

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Section: Operator-valued Measuresmentioning

confidence: 99%

Separability of the L¹-space of a vector measure

Ricker

1992

Glasgow Math. J.

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“…Conversely, if x' e X', then its restriction to P (2)[x] is continuous, and hence the right-hand side of (5), being equal to x'° <P Px , defines an element of L\Px) '.…”

Section: Proofs Of Theorems 3 Andmentioning

confidence: 99%

“…(ii) If x G X, then Px can be interpreted as having values in the cyclic space / > (2)[x] equipped with the relative A'-topology, and hence there is a finite positive measure X x on 2 such that Px is absolutely continuous with respect to X x [6, Proposition 4]. The conclusion then follows from an argument as in the proof of [13,Theorem 1].…”

Section: Criteria For Closedness Of Spectral Measuresmentioning

confidence: 99%

“…For example, let X denote the Hilbert space / 2 ([0,1]) and let P denote the \J^X)-valued spectral measure of multiplication by characteristic functions of Borel subsets of [0,1]. Then it is easily verified that each cyclic space P (2)[x], x G X, is separable (since the support of x is a countable subset of [0,1]). However, since the range of P is clearly not a closed subset of \-{X), it follows that P is not a closed measure [11,Proposition 3].…”

Section: Criteria For Closedness Of Spectral Measuresmentioning

confidence: 99%

“…The adjoint of an operator F e L ( J f ) is denoted by V. The correspondence E, * , . « * ; -» £ e (L( X))', defined by (2) f : ris an (algebraic) isomorphism of the tensor product X ® X' onto the dual of L( X). Call an element of (L( A"))' elementary if it is of the form for some x e X and x' e X', in which case we will denote it by x ® x'.…”

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A concrete realization of the dual space of L¹-spaces of certain vector and operator-valued measures

Ricker

1987

J Aust Math Soc A

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For a given vector measure n, an important problem, but in practice a difficult one, is to give a concrete description of the dual space of L l (n). In this note such a description is presented for an important class of measures n, namely the spectral measures (in the sense of N. Dunford) and certain other vector and operator-valued measures that they naturally induce. The basic idea is to represent the 1} -spaces of such measures as a more familiar space whose dual space is known.

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Spectra of Scalar‐Type Spectral Operators and Schauder Decompositions

Okada

1988

Mathematische Nachrichten

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I t is well known that every non-empty closed subset of the complex plane is the spectrum of a normal operator in a HUBERT space. This result cannot be extended to the case of BANACH spaces. Indeed, it is shown in [17] that the spectrum of a spectral operator in a GR~THENDIECX space with the DUNFORD-PETTIS property is a finite set; the argument is based on the fact that such a BANACH space has no SCHAUDER decomposition. The purpose of this note is to show that, in a locally convex spice, the existence of an unconditional SCHAUDER decomposition ensures the property that every non-empty closed subset of the complex plane is the spectrum of a scalar-type spectral operator (see Theorem 6). As a consequence, a BANACH space has the above property if and only if it has an unconditional SCHAUDER decomposition (see Corollary 7).Throughout this note, let X be a sequentially complete, locally convex &us-DORFP space and X' its dual space. Let L ( X ) denote the space of all continuous linear operators from X into itself. If the space L ( X ) is equipped with the topology of pointwise convergence on X, then it will be denoted by L , ( X ) . The identity operator on the space X will be denoted by I . Let T be a linear operator defined on a subspace of X with values in X. Suppose that 1 is a complex number such that the operator 1I -T is an injection with dense range whose inverse (U -T)-1 defined on the range of 1I -T is extendable to a continuous operator from X into itself. Then, the extension of (11-T)-1 to X will be denoted by R (1.; T). The resolvent equation R(1; T)-R(C; T ) = -( I -[ ) R ( I ; T ) R([; T )holds whenever I and i are complex numbers for which the operators R(A; T ) and R(C; T ) exist.The resolvent set e(T) of a linear operator T from a subspace of X into X consists of those complex numbers 1 such that there is a neighbourhood U of 1 in the

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Spectral measures and the Bade reflexivity theorem

Cited by 33 publications

References 11 publications

Separability of the L¹-space of a vector measure

Separability of the L¹-space of a vector measure

A concrete realization of the dual space of L¹-spaces of certain vector and operator-valued measures

Spectra of Scalar‐Type Spectral Operators and Schauder Decompositions

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Spectral measures and the Bade reflexivity theorem

Cited by 33 publications

References 11 publications

Separability of the L1-space of a vector measure

Separability of the L1-space of a vector measure

A concrete realization of the dual space of L1-spaces of certain vector and operator-valued measures

Spectra of Scalar‐Type Spectral Operators and Schauder Decompositions

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Product

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Separability of the L¹-space of a vector measure

Separability of the L¹-space of a vector measure

A concrete realization of the dual space of L¹-spaces of certain vector and operator-valued measures