Ranges of spectral measures and Boolean algebras of projections

1999 ◽  
pp. 57-66
Author(s):  
Werner Ricker
Keyword(s):  
Boolean Algebras ◽  
Spectral Measures

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 Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

Information storage and retrieval systems with incomplete information I

1979 ◽  
Vol 2 (1) ◽  
pp. 17-41
Author(s):  
Michał Jaegermann
Keyword(s):  
Incomplete Information ◽  
Information Storage And Retrieval ◽  
Boolean Algebras ◽  
Information Storage ◽  
Storage And Retrieval ◽  
Retrieval Systems ◽  
Theory Of Information

In the paper is developed a theory of information storage and retrieval systems which arise in situations when a whole possessed information amounts to a fact that a given document has some feature from properly chosen set. Such systems are described as suitable maps from descriptor algebras into sets of subsets of sets of documents. Since descriptor algebras turn out to be pseudo-Boolean algebras, hence an “inner logic” of our systems is intuitionistic. In the paper is given a construction of systems and are considered theirs properties. We will show also (in Part II) a formalized theory of such systems.

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