scholarly journals Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

1997 ◽  
Vol 40 (3) ◽  
pp. 425-435 ◽  
Author(s):  
B. de Pagter ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Boolean Algebras ◽  
Spectral Operator ◽  
Strong Operator ◽  
Scalar Type ◽  
Resolution Of The Identity ◽  
Separable Spaces ◽  
Operator Closure ◽  
Positive Results

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

scholarly journals On the Carleman classes of vectors of a scalar type spectral operator

2004 ◽  
Vol 2004 (60) ◽  
pp. 3219-3235 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Resolution Of The Identity ◽  
Wiener Type

The Carleman classes of a scalar type spectral operator in a reflexive Banach space are characterized in terms of the operator's resolution of the identity. A theorem of the Paley-Wiener type is considered as an application.

Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

2011 ◽  
Vol 54 (2) ◽  
pp. 515-529
Author(s):  
Philip G. Spain
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Boolean Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Weak Star ◽  
Weakly Closed ◽  
Operator Closure ◽  
Dual Banach Space ◽  
Completion Theorem

AbstractPalmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

2019 ◽  
Vol 17 (1) ◽  
pp. 1082-1112
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
A Priori ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

Uniform operator σ-additivity of indefinite integrals induced by scalar-type spectral operators

1985 ◽  
Vol 101 (1-2) ◽  
pp. 141-146 ◽  
Author(s):  
S. Okada ◽  
W. Ricker
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Borel Sets ◽  
Scalar Type ◽  
Resolution Of The Identity ◽  
Spectral Integral ◽  
Set Function

SynopsisThis note characterises those Banach space valued, scalar-type spectral operators T = ∫ z dP(z), where P is the resolution of the identity for T, whose indefinite spectral integral E→∫EzdP(z) as a set function of the Borel sets of the complex plane is countably additive with respect to the uniform operator topology.

scholarly journals On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiableC0-semigroups

2004 ◽  
Vol 2004 (45) ◽  
pp. 2401-2422 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Spectral Operator ◽  
Scalar Type ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a scalar type spectral operator in a Banach space to be a generator of an infinite differentiable or a Gevrey ultradifferentiableC0-semigroup are found, the latter formulated exclusively in terms of the operator's spectrum.

scholarly journals On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient ◽  
Space Conditions

For the evolution equation with a scalar type spectral operator in a Banach space, conditions on are found that are necessary and sufficient for all weak solutions of the equation on to be strongly infinite differentiable on or . Certain effects of smoothness improvement of the weak solutions are analyzed.

scholarly journals Comment on “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator”

2018 ◽  
Vol 2018 ◽  
pp. 1-3
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Gevrey Classes ◽  
Scalar Type

The results of three papers, in which the author inadvertently overlooks certain deficiencies in the descriptions of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a complex Banach space established in “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator,” Int. J. Math. Math. Sci. 2004 (2004), no. 60, 3219–3235, are observed to remain true due to more recent findings.

scholarly journals Spectral measures on spaces not containing l∞

1981 ◽  
Vol 24 (1) ◽  
pp. 41-45 ◽  
Author(s):  
T. A. Gillespie
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Spectral Measure ◽  
Dual Space ◽  
Boolean Algebras ◽  
Special Role ◽  
Spectral Measures ◽  
Sequential Completeness ◽  
Algebra Of Sets

The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that(iii) every prespectral operator on X is spectral.(See also (1, Theorem 6.11, p. 165) for (iii).)

scholarly journals The strong closure of Boolean algebras of projections in Banach spaces

2004 ◽  
Vol 77 (3) ◽  
pp. 365-370 ◽  
Author(s):  
J. Diestel ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Sequence Space ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Special Cases

AbstractThis note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

2019 ◽  
Vol 17 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation

Abstract It is shown that, if all weak solutions of the evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

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