Compact well-bounded operators

Qingping, Cheng; Doust, Ian

doi:10.1017/s0017089501030087

“…. , it follows from [CD2] that the second partial sum converges (in norm) and that B is well-bounded. The case of the first sum is just a little more delicate.…”

Section: Lemma Suppose That {Zmentioning

confidence: 96%

An example in the theory of $AC$-operators

Doust

¹

,

Gillespie

²

2000

Proc. Amer. Math. Soc.

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0

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Abstract. AC-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact AC-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper.

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“…Let P = Qn -Qn+l, Then {P } forms a sequence of disjoint finite-rank pro-1 jections so Theorem 2.3 [53], shows that T= En=1 -Pn is well-bounded. Indeed, n since Q, ti -+ 0 in the strong operator topology, the operator T is well-bounded of type (B).…”

Section: Proof ([65] Theorem 5) 11mentioning

confidence: 99%

Spectral Theory of Linear Operators

Dowson¹

2007

Operator Theory: Advances 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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“…1 jections so Theorem 2.3[53], shows that T= En=1-Pn is well-bounded. Indeed, n since Q, ti -+ 0 in the strong operator topology, the operator T is well-bounded of type (B).…”

mentioning

confidence: 93%

Spectral Theory of Linear Operators

Pugachev¹,

Sinitsyn²

1999

Lectures on Functional Analysis and Applications

0

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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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Compact well-bounded operators

Cited by 6 publications

References 14 publications

An example in the theory of $AC$-operators

An example in the theory of $AC$-operators

Spectral Theory of Linear Operators

Spectral Theory of Linear Operators

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